A quick way in your head is to use base multiplication. In this case base 20... 17 + 3 = 20 17 - 3 = 14 14 * 20 = 280 3 * 3 = 9 280 + 9 = 289 Or base 40 for 36² 36 + 4 = 40 36 - 4 = 32 40 * 32 = 1280 4 * 4 = 16 1280 + 16 = 1296 🤗
@@williambiggs3699 unfortunately that’s not how multiplication works. For example, 7 x 7 is 49, but 6 x 8 is 48, and 5 x 9 is 45, and 4 x 10 is 40, and 3 x 11 is 33, and 2 x 12 is 24, and 1 x 13 is 13, and 0 x 14 is 0, and -1 x 15 is -15, and -2 x 16 is -32, and it just keeps decreasing by more every time. If you don’t believe that, try 8 x 8 is 64, 7 x 9 is 63. This means that 17 x 17 DOES NOT EQUAL 14 x 20. Sorry about that.
@@grandpaobvious algorithms are a key part to how the (human) world sustains itself. They are part of every facet of our technological life; from your mcdonalds order and grocery stores to space x and mars rovers.
I like how the four categories from your proof also show up in the "easier" proof. Either the multiple of 4 is above or below the prime, and either the multiple of 3 is above or below the prime, giving four possibilities that directly correspond to your categories.
To be fair, 2 is the only even prime. It breaks a lot of rules. You can modify it and use a different equation, and see it works for inverted positive integers: 2^2=(1/8)(24)+1 3^2=(1/3)(24)+1
If you read the book, you know he calls them sub primes because they are prime by default and don't even have an opportunity to be divided by anything.
@@rosiefay7283 I think the point is that 5 is the first prime greater than 2x2 (the first compound number), though I don't remember reading that part of the book so I might be wrong.
*LULZ* So Yeah -- I thought I caught a mistake in your maths there Nimmin, but I double-checked what you wrote -- 3*3 isn't the same as 3^3, my bad. *BUT LOOK.* 2^3 = 8 ; 3^3 = 27; 5^3 = 125; 7^3 = 343 Or, 1/3 of 24; 24 + 3; (5*24) + 5; (14*24) + 7.... :D
@@MithunGaming it's a running joke (meme if you will) when a calculation/classification is a miss, recategorizing them as a "Parker square or equation" etc. instead of identifying it as a miss.
It is truly a parker square because if they're all a multiple of 24+1, surely they are all also a multiple of 2+1. It's more graceful cause you don't have to call 3 subprime. Whatever even number above 1 you pick, it'll always have some cut off point where a prime is too small for it to work. Unless you pick 1+1, which includes all the primes because it is literally the definition of primes. Matt is just defining prime numbers here in a very weird and unnecessary way..
@Urrcreavesh I never claimed that it has anything to do with squares because it doesn't. I'm referencing a meme about the parker square because it's used whenever Matt tries to do something clever which is unimpressive and doesn't work very well. I think this is such an occasion. Look it up, it's on TH-cam somewhere
@@e11eohe11e 2 and 3 ARE real primes: a number is prime when it is divisible by only 2 numbers, 1 and itself. 2 is divisible only by 1 and itself, thus fulfilling the criteria, there's no such things as real or non-real primes. The only other category related to primes that I can remember now is semiprimes, which you get when multiplying 2 primes together.
Man you stole it from a scientist whose name i cant remember .The real statement goes like this “there is nothing certain in this universe except death ,taxes and the second law of thermodynamics”
I am currently doing my degree in maths and one of the things we need to prove is that all primes squared (above 3) are one more than a multiple of 6, and I know how to prove it because of your video. love you matt!!!!
STEP 01: Make a RULE STEP 02: When u find any element not following RULE, simply call them exceptions. STEP 03: When u find infinite such exceptions, say it's a COROLLARY of the main RULE! Now, u R done!!!
There is a point to be made, though, that 2 and 3 are the only primes which are smaller than the lowest compound number (which is 4). So they are kind of special in that way.
A prettier (or at least quicker) version of the first proof: (6m ± 1)^2 = 36m^2 ± 12m + 1 = 12m(3m ± 1) + 1 and 12m(3m ± 1) is divisible by 24 as either m is even, or 3m ± 1 is even
yeah this is much better, cause at least you dont need to assume that we have 2m or 2m+1 which is not necessarily true, just that even or odd which is 100% true
@CogitoErgoCogitoSum because you can-t divide it by 1 and itself since it's the same. Also we can do it from truth by contradiction. Let's say we have a prime p p is divisable by p and 1. if 1 is a prime then it is the only prime thus since having 1 singular prime is useless 1 is not a prime
@@patricksalhany8787 this is circular reasoning. The fundamental theorem of arithmetics assumes 1 is not prime. Then, you can't prove it with the fundamental theorem of arithmetics. 1 is not prime, because we have defined prime numbers to be such that they have exactly 2 divisors. 1 has only one divisor, so it is not a prime number.
Ok, say i think i like the idea of 1 being prime. I put on my magic hat and make everyone use the definition of prime as A prime is any positive integer factorisable only with itself and 1. What, except the trivial loss of the fundamental theorem of algebra, (which i would like to restate as every number can be written as a unique, simplest possible prime factorisation. Why would it not work?) what breaks? Please enlighten me in how our naturalistic understanding of math (i don’t have any clues about the ground-floor of peano-arithmetic, only that it is how i usually count and use numbers).
@@An_Amazing_Login5036 I mean, mathematicians used to consider 1 as a prime number, but as number theory evolved it was generally agreed upon that it's easier to just say that it isn't one. That way you avoid constantly saying "every prime except for 1". Also primes are interesting solely because of the Fundamental Theorem of Arithmetic. You could say they were "invented" as part of the theorem. So it would be kinda counterproductive to say 1 is prime but then also make an exception for it in the theorem. In a sense, 1 is "too special" to be "just" a prime number... it's sort of a foundational concept that's _even more_ fundamental than primes.
You can do it with (6n+1)^2: 36n^2+12n+1 12(3n^2+n) If you remember that n^2 is odd if n is odd and even if n is even, then you can see that 3*odd+odd will be even and 3*even+even is also even. So, it’s 12(even) which is a multiple of 24. You could also just factor it as 12n(3n+1), and either n or 3n+1 has to be even since if n is odd, 3*odd+1 is even
Just to spell it out for fatsquirrel75 5 is seximal 5 (0 x 6 + 5) 7 is seximal 11 (1 x 6 + 1) 11 is seximal 15 (1 x 6 + 5) 13 is seximal 21 (2 x 6 + 1) 17 is seximal 25 (2 x 6 + 5) 19 is seximal 31 (3 x 6 + 1) and so on. I love it! Thank you, @@HBMmaster.
I watch a lot of mathy channels, this one, loads of sci show, 3 blue 1 brown etc.. Somehow it has taken until today for me to realize that though I love these videos, deep down, I come here for the comments section.
I paused at 9 seconds to work it out with algebra. It makes tons of sense! I knew right away that it was reasonable since prime numbers themselves have a similar multiple+offset pattern, where they are 6n+-1
Einstein used to carry a cheat sheet around with various fundamental constants written down and Ramanujan lost a mental-calculation contest to a random guy at Cambridge.
Some mathematicians love to make themselves appear all mighty and invincible, but they ALSO struggle with math every now and then. Like Matt Parker himself has said a few times - math nerds don't necessarily love math just because it's "easy", they love it because they enjoy its difficulty.
You can do it all at the same time: (6k +- 1)^2 = 36 k^2 +- 12k + 1 Then factor out the common stuff in the first two terms: = 12k(3k +- 1) + 1 Either k is even, or, if k is odd, then (3k +- 1) is even. In either case, 12k(3k +- 1) is a multiple of 24.
Here's an algebraic simpler version: (6k +/- 1)^2=36k^2 +/- 12k + 1 Rearange to: 24k^2 + 12(k^2 +/- k) +1 = 24k^2 + 12(k(k +/- 1)) + 1 Now, either k or k+/-1 is even so we can write : 24k^2 + 24(k(k +/- 1)/2) + 1 = 24(k^2 + k(k +/- 1)/2) + 1 = 24N +1, where N must be an integer since both k^2 and k(k +/- 1)/2 are. QED
I did something midway between yours and the one in the video: (6k +/- 1)^2=36k^2 +/- 12k + 1 = 12k(3k +- 1) +1. Since k(3k +-1) is divisible by 2 as either k or (3k +-1) must be, then 12k(3k +-1) must be divisible by 24.
There's a far easier method. The squares of 1 3 5 7 mod 8 are all 1. And the squares of 1 2 mod 3 are all 1. Combine the two and it must be one more than a multiple of 24.
Akshat Aggarwal I literally said in my first comment that 2 and 3 were prime numbers. I was explaining that one of the reasons why he didn’t include them was because the proof doesn’t allow for the numbers to be divisible by 2 or 3.
Actually it is possible to prove that a multiple of 6 +- 1 has rest 1 in the division by 24. x = (6k+-1)^2 mod 24 x = 36k^2 +- 12k + 1 mod 24 x = 12k^2 +- 12k +1 mod 24 x = 12 * k*(k +- 1) + 1 mod 24 And since k*(k +- 1)=0 mod 2, because it is the product of two consecutive integers (and therefore must be even) x = 1 mod 24
I think the word for the second proof is "elegant", it's compact, gets the job done. But elegance in design often comes after the working out and pruning of things that are unnecessary, and are often not the route that is taken by a pathfinder; instead, it's the shortest route that you can really only clearly see after you've made it to the destination. I always think of when I would be off-trail in the mountains and come across something interesting. The path I would take people on to come and view the interesting thing was usually much shorter than the route I took to discover it, because now I have the destination and you can find the "shortest route" to it. I think the mental path of discovery is very analogous, and I'm happy that Matt has made a point of showing the more circuitous paths, I think it really makes the journey seem more accessible to people and de-mystifies math and knowledge, which is all too often held up as unattainable and some sort of magic. Yea, once you point something out to other agents and experts in your space, people will start optimizing immediately, and the result of that peer-engagement usually has that sort of elegant and beautiful quality. But, often the most innovative ideas come from a mind that is just bent on finding "A" better way or "A" solution, and it's great to showcase that grit and brute-force and inelegance are not enemies of furthering understanding and knowledge, while at the same time, showing how engagement with other experts takes a "cool" idea, and turns it into something beautiful. --- Thanks Matt (If you're still reading comments on here 4 years later)
Great video. I've always been fascinated with primes. The first thing I did when I got my forst computer(a Commodore 64 (khz processor speed) was to write a prime number generator and then tweak it until it would run really fast. Gees, what a geek.
You might like Dave's Garage channel. He talks a lot about programming prime number finders as a kid on very early computers and optimizing the code and now he uses the same code to test the speeds between 100 different programming languages.
I was writing a program to check if a number is prime or not and I used this mathematical concept over there. I just realized that though 2 and 3 don't fit into Matt's theory, but they can be applied to the concept in reverse manner, i.e, (2*2 -1) and (3*3-1) divide 24 perfectly. That helped in optimization of my solution.
Aaron P.. They are real primes, but different from all the others. There is no way a non prime number can be in between 1 and 2 or 1 and 3, so it's a bit obvious that 2 and 3 must be prime. 5 is the first prime that has a non prime between it and 1 (namely 4)
Hey Matt, it's a way shorter to show that (6n+1)² or (6n-1)² are Multiples of 24 plus 1 For Example (6n+1)² = 36n²+12n+1 = 12( 3n²+n) +1 3n²+n is always a Even number because if n is uneven you have 3*uneven²+uneven which alswes ends up beeing even because uneven+uneven = even and if n is even you have 3*even²+even which is even, too Therefore there is always a k from the natural numbers such that 3n²+n = 2k With that you have 12( 3n²+n) +1 = 12*(2k)+1 = 24k+1 You can do the same with (6n-1)²
I prefer factoring to 12n(3n±1) + 1. For 12n(3n±1) to be a multiple of 24, you need n or (3n±1) to be even. If n is even, we're done. If n is odd, then 3n is odd, and adding or subtracting 1 gives an even number, so (3n±1) is even.
@@EnteiFire4 you can also use the p=6 plus or minus 1 fact, and note that of p-1 and p+1 in the factorization p^2-1=(p-1)(p+1), one is going to be a multiple of 6 and the other a multiple of 6 plus or minus 2 and so is a multiple of 4.
@@richardfredlund3802 I like that! Although I think the pair is either a multiple of 6 and a multiple of 4, *or* a multiple of 12 and a multiple of 2. That still works, though.
To put it another way, the product of any two consecutive even numbers is a multiple of 8. So the square of any odd number is one more than a multiple of 8. And since all primes past 2 are odd, all you need is that one of those factors is a multiple of 3.
Or more concisely: If 2 doesn't divide p, 8 divides p^2-1. If 3 doesn't divide p, 3 divides p^2-1. So if neither 2 nor 3 divide p, then 24 divides p^2-1.
2 and 3 work too. 2^2 is (24 * 1/8 + 1), and 3^2 is (24* 1/3 + 1). And since the multiplier is a fraction less than 1, I am with Matt on calling these two numbers as sub-prime.
I'm sure someone somewhere said this (and I haven't finished watching the video, so maybe they'll cover it?) but 2^2 - 1 = 3, and 3^2 - 1 = 8 .... and 3 x 8 is 24 :)
My teacher actually had me and his other students prove this on a test. He expected us to use equivalence classes in mod 24. The proof follows these steps: 1) Partition the set of all integers by all of the equivalence classes in mod 24. 2) Consider the classes as the range of numbers from -11 to 12 (these numbers are actually equivalence classes, so they represent the set of all integers). 3) Cross out all of the multiples of two and all of the multiples of three. (We’re left with the equivalence classes -11, -7, -5, -1, 1, 5, 7, and 11, all still in mod 24). 4) Square each number and minus one. The new numbers are 0, 24, 48, and 120, which are all multiples of 24. Of course, this proof does not show that only primes have this property. It only shows that numbers which are not multiples of two or three have this property, and since all primes are not multiples of two or three, they have this property. So, there are definitely numbers that aren’t multiples of two or three but are not prime, just like Matt showed in the video (e.g. 25). Such numbers are those of which there are multiple prime factors and none of the prime factors are two or three. In the case of 25, its prime factorization is 5 and 5, so it is one of the numbers that is not a multiple of two or three and is not a prime number. But it is definitely true that prime numbers are not multiples of two or three, so they can be squared and end up being one more than a multiple of 24.
I've been interested in and studies prime numbers since I was 14 years old, and next month I will be 74, so that's 60 years. I've found all sorts of interesting, quirky facts about them. They are some of the most fascinating numbers to study, because it seems like there should be no patterns and yet they are everywhere.
Using the same method as the second (more creative) proof, it also turns out that if you take the square of a prime number and multiply it by that same square minus five, you'll always end up with four less than a multiple of 360. Example (using the prime number 7): 49 × 41 = 2156 = 2160 - 4, and 2160 = 360 × 6. The proof comes from multiplying the factors (p - 2) (p - 1) (p + 1) and (p + 2). You'd end up with a polynomial that looks like p^4 - 5p^2 + 4, which can be rewritten as p^2 (p^2 - 5) + 4. When you look at the four factors on a number line, in addition to having a multiple of 2, 3, and 4, the newly added (p - 2) and (p + 2) also guarantee a second multiple of 3 as well as a multiple of 5 (but only if you're using prime numbers higher than 5). Therefore, since 2 × 3 × 3 × 4 × 5 = 360, you can guarantee that multiplying all four factors will give you a multiple of 360.
I had no idea the primes could be divided into categories like this! In my (admittedly limited) maths education I got the impression that the defining characteristic is being absolutely without patterns. This video, as well as another video where you actually directly state that primes do have patterns, have enlightened me! Thank you. :)
Its not a generator, because not every (24k + 1 ) is prime. So its really not showing a pattern. Its created a pattern for possible primes, just the same as "not even" creates a pattern for possible primes. Now show a pattern to ALL the primes and ONLY the primes.
24 used to be my favorite number. Many of the reasons why it was my favorite number is basically the same reason why some people suggest Dozenal is a better number system than Decimal, it just divides nicely by a lot of single digit numbers.
This is just a really easy number theory problem. We just use the fact that all primes can be written as 6k+-1 excluding 2 and 3. This fact is simple: We could have 6k,6k+1,6k+2,6k+3,6k+4,6k+5. Unless the prime is 2 or 3, we must have P=6k+5 or 6k+1. 6k+5 is the same as a number of the form 6k-1. So we're just squaring 6k+-1. We just get P^2= 12(3k^2+-k)+1 3k^2-k is the same as (3k-1)k. Either k or 3k-1 will be even (If k is odd 3k-1 is even and if k is even k is even). Then 3k^2+-k is the same as 2n for some integer n. Plugging in gives us P^2=12(2n)+1=24n+1 with the exceptions of 2 and 3. This isn't special about primes- Any number of the form (6k+-1)^2 is one more than a multiple of 24.
I'm definitely a Matt Parker type of maths enthusiast. I love maths, and I really appreciate the beauty of that second proof, but I would've for sure gone down the route of the first proof if I was solving this. I wish I had the intuition to solve problems the way the second proof does, but I don't.
Strange but true. Proof by cases can be very helpful. It’s also why most mathematicians do their best work while they are young and creative. The genius of many mathematicians comes from clever ways to rethink of problems in (relatively) simpler terms
Very interesting. I already knew that 17^2 is 289 because, well, I like numbers, especially primes, and just happened to know that. Incidentally, genius savant Daniel Tammet called 289 an "ugly" number (in his incredible synesthetic mind), but I find the number 289 quite lovely.
You can do it from the 6k+1 and 6k-1 cases. Squaring 6k+1 gives 36kk + 12k + 1, which is 24(1.5kk + 0.5k) + 1. Squaring 6k-1 gives 36kk - 12k + 1, which is 24(1.5kk - 0.5k) + 1. If k is odd then k squared is odd, if k is even, then k squared is even - therefore the bit in brackets is an integer. QED.
I had a geometry professor in community college always say, "Matthew, make this proof more elegant." At the time I didn't know what he meant. It wasn't until my capstone math course that I finally got what he meant. No other professor ever said it. I have my bachelors in math now. I'm with you. When he said easier, I immediately thought, nah that's more elegant.
@@Thedeadbeatmatt tbf, what is "easier" depends on where you're coming from at the moment. For me, the whole proposition seemed almost trivial and the p^2 - 1 approach sounded very similar to something I would try first. But that is because of something I have been working on that is actually very related to that, so of course I would try something more like it (that likely would quickly reduce to it itself).
Elegant proofs when clearly explained are usually more understandable. The brute force approach is arguably a stronger demonstration of primes occur next to 6. The elegant version requires the explanation to follow.
Want an even more elegant one? All primes are +-1 mod 3, which means all prime number squares are 1 mod 3. All primes are +-1 or +-3 mod 8 which means all prime number squares are 1 or 9 mod 8, and 9 is also 1 mod 8. Combine those two facts to get that all prime number squares are 1 mod 24.
My approach (which is very close to the (p+1)*(p-1) explanation: 1) Every prime number can be either expressed by 3a +1 or by 3a + 2. (a is an integer) (3a+1)² = 9a² + 6a + 1 -> (3a+1)² - 1 can be divided by 3 (3a+2)² = 9a² + 12a + 4 = 9a² + 12a + 3 +1 -> (3a+2)² - 1 can be divided by 3 2) Every prime number can be expressed as 2b+1 (2b+1)² = 4b² + 4b + 1 -> if b=2c (i.e. b is even), then (2b+1)² = 16c² + 8c + 1 -> (2b+1)²-1 can be divided by 8 if b is even -> if b=2c+1 (i.e. b is odd), then (2b+1)² = (4c+3)² = 16c² + 24c + 9 = 16c² + 24 + 8 + 1 -> (2b+1)²-1 can be divided by 8 if b is odd. -> p²-1 can be divided by 8 and by 3 and therfore by 24...
Even easier: Ignoring 2 and 3, a prime is either 6m+1 or 6m-1. (6m+1)^2= 36m^2+12m+1 = 12m(3m+1)+1. If m is odd, m = 2k+1. 3(2k+1)+1=6k+3+1=6k+4=2(3k+2) which is divisible by 2 so you can factor out another 2 to get 24m(3k+2) + 1. If m is even, m=2k which means 12m=12(2k)=24k. So (6m+1)^2 is either 24m(3k+2)+1 or 24k(3m+1)+1. It works the same way if the prime is 6m-1.
Let k be integer and p(n) be the n-th prime number, then: p(n>2)^2-1 = 1 x 24 x k p(n>3)^4-1 = 10 x 24 x k p(n>4)^6-1 = 21 x 24 x k p(n>3)^8-1 = 20 x 24 x k p(n>5)^10-1 = 11 x 24 x k p(n>3)^12-1 = 2730 x 24 x k p(n>2)^14-1 = 1 x 24 x k p(n>7)^16-1 = 680 x 24 x k p(n>8)^18-1 = 1197 x 24 x k p(n>5)^20-1 = 550 x 24 x k p(n>9)^22-1 = 23 x 24 x k p(n>6)^24-1 = 5460 x 24 x k ... As usual, the 24th power is a show off...
Whenever he was trying to compute 17^2 and was coming up with an easy way to do it, I immediately thought "that's gonna be 170 times 2, minus 3 lots of 17." I even paused the video and heard it in Matt's voice in my head. "170 times 2 is 340, 3 lots of 17, 51, 340 minus 51........ 289." You can hear it in his voice now, can't you?
I have been doing something similar as a easy trick to multiply squared numbers in my head. The difference of squares thing can be generalized. So, a^2, can be modified to a^2 - s^2 and it be changed to (a-s)(a+s). To solve for a^2, just add the s^2 back on to the answer. So, 19^2, can be rewritten as (19-1)(19+1) + 1 (or 361). 22^2 can be rewritten as (25)(19) + 9 or 484.
Open question: I’m from Canada and when we talk about mathematics we shorten it to “math” not “maths” the way you do in UK, Aus, etc. Any reason why 4:28 said “Math-related items” vs “maths” despite Matt and Brady’s Aus backgrounds? Am I up too late again?
Placing 6n+-1 in several rows on a spreadsheet, 1 thru 6, 7 thru 12, 13 thru 18 etc. The result here is that two columns will contain prime numbers. Now 2 * 3 * 5 = 30. Place this on the spreadsheet 1 thru 30, 31 thru 60, 61 thru 90 etc. The result here is that 8 columns will contain prime numbers. Now 2 * 3 * 5 * 7 = 210. Place this on the spreadsheet 1 thru 210, 211 thru 420, 421 thru 630, etc. The result here is that 43 columns will contain prime numbers. If we continue with this 2310 gives 340 columns of primes, 30030 gives 3243 columns of primes. This continues with each prime number in the factorial. This also indicates the possible location of the different size gaps.
I think it is indeed worth emphasizing that the "prime" part of the statement is basically a red herring. And that's something I say as a number theorist! The claim is just about integers which aren't divisible by 2 or 3, i.e., numbers which are coprime to 24. In the world of abstract algebra, we would say that the (multiplicative) group (Z/24Z)* is isomorphic to the (additive) group (Z/2Z)x(Z/2Z)x(Z/2Z). In practical terms, that implies that if n is an integer which is coprime to 24, then n^2 is congruent to 1 mod 24. Similarly, if you take any integer n which isn't divisible by 2, 3, 7 or 31, then n^(30) = 1 mod 5208. This is because: (Z/5208Z)* = (Z/2Z)^5 x (Z/3Z) x (Z/5Z). You can do similar things with 5208 replaced by any integer! You just need to look at the group structure. The only thing that makes some cases look special is finding an integer (like 24 in the video, or 5208 above) where the elements of the multiplicative group of units have small order compared to the size of the group. You can do this by finding a collection of primes p where p-1 is a "smooth" number.
Thank you for this very much for this video. The video shows that pi is approximately 22 / 7. This value is approximately 3.14. Using the properties of this value we can compute prime numbers in sequence, which is based on the existing computing capability. The formula was an algorithm, that was developed by a well known mathematician. Using his formula and the method that I discovered, I can compute prime numbers in sequence using 22 / 7 .
Easy way to find 17 squared: add 3 to get 20, subtract 3 to get 14, multiply 20 by 14 and add 3 squared. (a+b)(a-b) = a*a - b*b, thus (a+b)(a-b) + b*b = a*a Generally: choose smallest b so that a+b or a-b is a multiple of 10
TECHNICALLY there is not any pattern. But when we talk about patterns, we mean patterns like the Fibonacci sequence, 1, 1, 2, 3, 5, 8... and so on. Primes doesn't have an algebraic formula to follow, but there are ways to figure out the nth prime number in other ways. It really comes down to how you define a pattern though, which is a problem since everyone has unique definitions for just that.
2 squared is 4 4-1 is 3 3 is 24×(1/8) Not a *whole number* multiple of 24, but close. 3 squared is 9 9-1 is 8 8 is 24*(1/3) Not a *whole number* multiple, but close.
Every prime that is not 3 gives a remainder 1 or 2 (mod 3). It is easy to show that the squares of numbers which give a remainder either 1 or 2 (mod 3) always give remainder 1 (mod 3). Also since every prime except 2 is an odd number, it is also easy to show that a square of any odd number always gives a remainder of 1 (mod 8). At the end, since 3 and 8 are relativley prime we can say that squares of prime numbers bigger than 3 always give remainder of 1 (mod 24) which is equivalent to saying that the squares are in the form 24k+1
The powers of prime numbers and the numbers that can only be divided by the different prime numbers are very common. For example: 2^x, 6^x, 30^x, 210^x, etc.
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Tysm
17²=139.... the Parker Prime Square
The appropriate phrase is "more elegant".
7*7=49 7*1+1*7=14 14+4=18 1*1=1 1+1=2
17*17=289
Hi, can we hear more about sub-primes numbers, is there really a way to define them?
Watching a PhD mathematician struggle to get 17^2 was reassuring.
It was a Parker square...
does he have a PhD?
A quick way in your head is to use base multiplication.
In this case base 20...
17 + 3 = 20
17 - 3 = 14
14 * 20 = 280
3 * 3 = 9
280 + 9 = 289
Or base 40 for 36²
36 + 4 = 40
36 - 4 = 32
40 * 32 = 1280
4 * 4 = 16
1280 + 16 = 1296 🤗
Ah, binomial theorem, my old nemesis, we meet again.
@@williambiggs3699 unfortunately that’s not how multiplication works.
For example, 7 x 7 is 49, but 6 x 8 is 48, and 5 x 9 is 45, and 4 x 10 is 40, and 3 x 11 is 33, and 2 x 12 is 24, and 1 x 13 is 13, and 0 x 14 is 0, and -1 x 15 is -15, and -2 x 16 is -32, and it just keeps decreasing by more every time.
If you don’t believe that, try 8 x 8 is 64, 7 x 9 is 63. This means that 17 x 17 DOES NOT EQUAL 14 x 20. Sorry about that.
0:58 I love how he rewrote the 139 to make it read 289 after he scored out that calculation so he could say "Dammit I was right". Parker convincing.
No of likes 468 . Divide it by 2 you get 234. Well now you have increased his like count
The sum was actually 289. Check it again
He wrote the 170 so it looked like a 110 that is why his maths is wrong. And he carried the 1 wrong.
Hello Richard.
That was a real parker square of a calculation.
Video starts with Matt trying to Parker square 17.
17 Parker squared
You're my hero.
Algorithms are for computer nerds.
and more
@@grandpaobvious algorithms are a key part to how the (human) world sustains itself. They are part of every facet of our technological life; from your mcdonalds order and grocery stores to space x and mars rovers.
I like how hard it was for him to do the math in his head. It reminds me of the saying "the more math you know, the harder it is to do math"
Who said that?
@@KemonoFren Joe
mental arithmetic is not "doing math" and you completely missed what that saying is, well, saying
@@GroovingPict Aye that’s the fun part about sayings, they gain power as both their original meaning and its inverse over time
knowing how to do math, and actually doing math are two different things.
I like how the four categories from your proof also show up in the "easier" proof. Either the multiple of 4 is above or below the prime, and either the multiple of 3 is above or below the prime, giving four possibilities that directly correspond to your categories.
It amazing how matt did all the mental maths perfectly and then said 170 +70 AND 49 is somehow less than the original
*Sees first two primes don’t follow his rule
*Calls them sub primes.
To be fair, 2 is the only even prime. It breaks a lot of rules.
You can modify it and use a different equation, and see it works for inverted positive integers:
2^2=(1/8)(24)+1
3^2=(1/3)(24)+1
If you read the book, you know he calls them sub primes because they are prime by default and don't even have an opportunity to be divided by anything.
@@nimmin8094 evenness is a poor property to use.
@@mrmimeisfunny *No* prime can be divided by anything -- if it could, it wouldn't be prime. [By any positive integer except itself and 1, of course.]
@@rosiefay7283 I think the point is that 5 is the first prime greater than 2x2 (the first compound number), though I don't remember reading that part of the book so I might be wrong.
17^2=139
I think I just witnessed Parker Squaring.
Underrated comment
4:31 Matt received weapons of math instruction
Said Mike Tyson
One of the few cases when the British “maths” (🤮) would be better because “maths” sounds more like “mass”
@@52flyingbicycles so you could have maths confusion and maths debate...?
That second proof gave me chills down my spine when I saw where it was going, you might even say it was an Amazingly Satisfying Mathematical Result
3*3 is one more than 8, 1/3 of 24.
2*2 is one more than 3, 1/8 of 24.
Pretty neat!
exactly, he didn't say that it had to be a whole number integer multiple of 24
Hahahah Slartbarg, in that case, all numbers are multiples of 24
*LULZ* So Yeah -- I thought I caught a mistake in your maths there Nimmin, but I double-checked what you wrote -- 3*3 isn't the same as 3^3, my bad.
*BUT LOOK.*
2^3 = 8 ; 3^3 = 27; 5^3 = 125; 7^3 = 343
Or, 1/3 of 24; 24 + 3; (5*24) + 5; (14*24) + 7.... :D
@@TheDGomezzi I was just going by inversions of integers
@@jamespfp My brains a bit slow this morning. I'm interested! I'll have a proper look this afternoon :)
17²=139, nice to start the video with a parker equation!
can someone make this T-shirt please?
Can you explain what a Parker equation is? Please
@@MithunGaming it's a running joke (meme if you will) when a calculation/classification is a miss, recategorizing them as a "Parker square or equation" etc. instead of identifying it as a miss.
Mithun Gaming - Parker Square or Parker Equation: A joke that has outlived it’s humor and should die.
the easiest way to work it out for me is just work out (17*20)-(17*3) and 17*10 is 170 which is above the 139 he worked out
Matt: There is a pattern with prime squares, they are all multiples of 24 plus 1.
All?
Matt: Almost all.
So it's a parker squares pattern then...
It is truly a parker square because if they're all a multiple of 24+1, surely they are all also a multiple of 2+1. It's more graceful cause you don't have to call 3 subprime. Whatever even number above 1 you pick, it'll always have some cut off point where a prime is too small for it to work. Unless you pick 1+1, which includes all the primes because it is literally the definition of primes. Matt is just defining prime numbers here in a very weird and unnecessary way..
@Urrcreavesh I never claimed that it has anything to do with squares because it doesn't. I'm referencing a meme about the parker square because it's used whenever Matt tries to do something clever which is unimpressive and doesn't work very well. I think this is such an occasion. Look it up, it's on TH-cam somewhere
@@alexvandenbroek5587 multiples of 2 + 1 does not sound impressive at all since that is the definition of an odd number.
Many primes do not follow his rule not just 2 and 3.
@@dannygjk example?
Nice sneaky edit of the 139 on the paper
I just love your guests, every one of them. Listening to them is such a treat.
Thanks
I sure hope the maths related items are for review and unboxing purposes.
You have more likes than one of Numberphile's pinned comments.
no hate, but i don't get unbox excitement and i'm jealous
@@skeletonrowdie1768 I generally agree, but calculator unboxing is a whole different beast.
MathSSSS
They could just for his store though
“I like to argue that 2 and 3 are not real primes”
Goodbye, fundamental theorem of arithmetic
is it iconoclasm or nihilism? We report, you decide!
CogitoErgoCogitoSum Its called a joke lol
@@quaternaryyy I am pretty sure he was joking too so... double r/wooosh for you, I guess.
welniok There is one reason to prefer Fahrenheit: Compared to Celsius, you usually get much more degrees in Fahrenheit.
@@e11eohe11e 2 and 3 ARE real primes: a number is prime when it is divisible by only 2 numbers, 1 and itself. 2 is divisible only by 1 and itself, thus fulfilling the criteria, there's no such things as real or non-real primes. The only other category related to primes that I can remember now is semiprimes, which you get when multiplying 2 primes together.
Only three things are certain: Death, Taxes, and Parker Square jokes.
Don't forget rice pudding.
Parker Squares from the Nile; does anybody else get the second reference?
Man you stole it from a scientist whose name i cant remember .The real statement goes like this “there is nothing certain in this universe except death ,taxes and the second law of thermodynamics”
Priyanshu Sudhakar No, the real statement is from Benjamin Franklin and it’s only the first two.
The fourth one are Uranus jokes. 😜😜😜😜
I am currently doing my degree in maths and one of the things we need to prove is that all primes squared (above 3) are one more than a multiple of 6, and I know how to prove it because of your video. love you matt!!!!
STEP 01: Make a RULE
STEP 02: When u find any element not following RULE, simply call them exceptions.
STEP 03: When u find infinite such exceptions, say it's a COROLLARY of the main RULE!
Now, u R done!!!
There is a point to be made, though, that 2 and 3 are the only primes which are smaller than the lowest compound number (which is 4). So they are kind of special in that way.
Any number plus half of itself is odd.
@ABHINAV JAIN That's an exeption.
@ABHINAV JAIN That's just a corollary of the main thing.
@ABHINAV JAIN that's the joke
17^2=139.
C'mon. Now you're just begging us to make a Parker Square joke.
And the way he did that, wth. (a+b)^2=a^2+2ab+b^2 is much easier to square numbers!
@@masansr yep. 20 * 14 = 280 and then + 3^2 = 289
Because basically you have (n+3)(n-3) = n^2 - 9. Then you just add 9 to get n^2
139 dislikes on the video...
17 x 17 is the easiest way, lol
@@mateuszm7882 yeah haha
A prettier (or at least quicker) version of the first proof:
(6m ± 1)^2 = 36m^2 ± 12m + 1
= 12m(3m ± 1) + 1
and 12m(3m ± 1) is divisible by 24 as either m is even, or 3m ± 1 is even
Just posted similar comment - yours is better.
Also just posted a similar comment, yours is better since I don’t know how to do +/- without copy and pasting it online which I was too lazy to do.
yeah this is much better, cause at least you dont need to assume that we have 2m or 2m+1 which is not necessarily true, just that even or odd which is 100% true
@@krowa1010 Um, even numbers can all be written as 2m, and all odd numbers can be written 2m+1. What are you trying to say?
2 and 3 are not primes but subprimes?
Mmmmm
I too like to live dangerously.
@CogitoErgoCogitoSum because you can-t divide it by 1 and itself since it's the same. Also we can do it from truth by contradiction. Let's say we have a prime p p is divisable by p and 1. if 1 is a prime then it is the only prime thus since having 1 singular prime is useless 1 is not a prime
@@patricksalhany8787 this is circular reasoning. The fundamental theorem of arithmetics assumes 1 is not prime. Then, you can't prove it with the fundamental theorem of arithmetics.
1 is not prime, because we have defined prime numbers to be such that they have exactly 2 divisors. 1 has only one divisor, so it is not a prime number.
@CogitoErgoCogitoSum Because we define them to have exactly 2 divisors: 1 and themselves
Ok, say i think i like the idea of 1 being prime. I put on my magic hat and make everyone use the definition of prime as
A prime is any positive integer factorisable only with itself and 1.
What, except the trivial loss of the fundamental theorem of algebra, (which i would like to restate as every number can be written as a unique, simplest possible prime factorisation. Why would it not work?) what breaks? Please enlighten me in how our naturalistic understanding of math (i don’t have any clues about the ground-floor of peano-arithmetic, only that it is how i usually count and use numbers).
@@An_Amazing_Login5036 I mean, mathematicians used to consider 1 as a prime number, but as number theory evolved it was generally agreed upon that it's easier to just say that it isn't one. That way you avoid constantly saying "every prime except for 1". Also primes are interesting solely because of the Fundamental Theorem of Arithmetic. You could say they were "invented" as part of the theorem. So it would be kinda counterproductive to say 1 is prime but then also make an exception for it in the theorem.
In a sense, 1 is "too special" to be "just" a prime number... it's sort of a foundational concept that's _even more_ fundamental than primes.
my theorem: every prime cubed is one more than a multiple of 2.
Well yes. A prime will be an odd number. The product of 3 odd numbers will be odd. Was this a joke? I'm dim.
of course it's a joke @@ronindebeatrice
yair koskas wrong.
2 is a prime.
2^3=8.
8 is not 1 more than a multiple of 2.
Your theorem is wrong.
@@patricksalhany8787 so this is the only prime that doesn't follow my theorem
@@MrYairosh yeah, but you said EVERY prime, so including 2.
You can do it with (6n+1)^2:
36n^2+12n+1
12(3n^2+n)
If you remember that n^2 is odd if n is odd and even if n is even, then you can see that 3*odd+odd will be even and 3*even+even is also even. So, it’s 12(even) which is a multiple of 24.
You could also just factor it as
12n(3n+1), and either n or 3n+1 has to be even since if n is odd, 3*odd+1 is even
lol i love how he failed 17^2
What's weird is that he said that 17^2 is 170 plus something, but he got at the end 139 which is less than 170.
Aliens.
Parker square!
@diego maradonna I thought you were only a footballer, but you also do maths.
Wow!
Keep up the great work dude !
@diego maradonna ohhhhh. That is sad.
And the dude has a PhD in mathematics!
every prime being adjacent to a multiple of six is yet another reason why seximal is the best numbering system (all primes end with 1 or 5!)
Fails at 2 and 3
@@galoomba5559 I prefer the unary number system, since every prime including 2 ends in 1.
@@galoomba5559 correct
@@effuah Every prime adjacent to a multiple of six does not include 2 and 3.
Just to spell it out for fatsquirrel75
5 is seximal 5 (0 x 6 + 5)
7 is seximal 11 (1 x 6 + 1)
11 is seximal 15 (1 x 6 + 5)
13 is seximal 21 (2 x 6 + 1)
17 is seximal 25 (2 x 6 + 5)
19 is seximal 31 (3 x 6 + 1)
and so on.
I love it! Thank you, @@HBMmaster.
I watch a lot of mathy channels, this one, loads of sci show, 3 blue 1 brown etc.. Somehow it has taken until today for me to realize that though I love these videos, deep down, I come here for the comments section.
P^2 - 1 way of proving is so very elegant. It really melts my heart. Simple & Brilliant
91 = 24*345 + 1, but 91 is not a prime :)
@@abhinavs2484 91 is not a square either. 91 = 7 x 13
So Matt's method, in its inferiority, could be called "The Parker Proof".
Right, but someone earlier called it the Parker goof.
Many primes do not follow that rule not just 2 and 3.
@@dannygjk like which one?
I paused at 9 seconds to work it out with algebra. It makes tons of sense! I knew right away that it was reasonable since prime numbers themselves have a similar multiple+offset pattern, where they are 6n+-1
8:15 "I did this way. This is mine. I love it."
That's the sipirit!
of a classic Parker Squarer.
Keep calm and square on.
Makes me feel human even mathematicians trouble with head calculations!
Einstein used to carry a cheat sheet around with various fundamental constants written down and Ramanujan lost a mental-calculation contest to a random guy at Cambridge.
The professor who got me to understand calculus couldn't tie his shoes
Charles M - I’m a successful biz man and can’t tie a necktie.
We all have our strengths & weaknesses
Nah it's just a Parker Square he meant to do that.
Some mathematicians love to make themselves appear all mighty and invincible, but they ALSO struggle with math every now and then.
Like Matt Parker himself has said a few times - math nerds don't necessarily love math just because it's "easy", they love it because they enjoy its difficulty.
Wow, the second demonstration is very clever. I wouldn't have found it
> supposed to work for all primes
> works for almost all (not 2 and 3)
> the parker square of prime patterns
They aren't subprimes, they are Parker Primes :)
upvote that man
This comment is under appreciated.
You can do it all at the same time:
(6k +- 1)^2 = 36 k^2 +- 12k + 1
Then factor out the common stuff in the first two terms:
= 12k(3k +- 1) + 1
Either k is even, or, if k is odd, then (3k +- 1) is even.
In either case, 12k(3k +- 1) is a multiple of 24.
I did about the same. A lot simpler than his four cases.
Here's an algebraic simpler version:
(6k +/- 1)^2=36k^2 +/- 12k + 1
Rearange to:
24k^2 + 12(k^2 +/- k) +1 = 24k^2 + 12(k(k +/- 1)) + 1
Now, either k or k+/-1 is even so we can write :
24k^2 + 24(k(k +/- 1)/2) + 1 = 24(k^2 + k(k +/- 1)/2) + 1 = 24N +1, where N must be an integer since both k^2 and k(k +/- 1)/2 are.
QED
I did something midway between yours and the one in the video: (6k +/- 1)^2=36k^2 +/- 12k + 1 = 12k(3k +- 1) +1. Since k(3k +-1) is divisible by 2 as either k or (3k +-1) must be, then 12k(3k +-1) must be divisible by 24.
There's a far easier method. The squares of 1 3 5 7 mod 8 are all 1. And the squares of 1 2 mod 3 are all 1. Combine the two and it must be one more than a multiple of 24.
So 139 is the Parker square of 17, huh.
DrSnap23 underated comment
Does that mean 2 and 3 are Parker primes?
I love numbers theory, esp. with primes! So amazing and easy to follow! Keep it on!
The fact that you don't count 2 and 3 as proper prime numbers is the REAL subprime crisis.
S D to be fair, both 2 and 3 are the only prime numbers divisible by 2 and 3 respectively
@@haidynwendlandt2479 Dude,do u even know the definition of prime numbers?
Akshat Aggarwal The proof forces the numbers not be divisible by 2 or 3, so every prime number greater than 3 works
@@haidynwendlandt2479 that doesn't explain ur 1st comment,it doesn't make any sense.
Akshat Aggarwal I literally said in my first comment that 2 and 3 were prime numbers. I was explaining that one of the reasons why he didn’t include them was because the proof doesn’t allow for the numbers to be divisible by 2 or 3.
Actually it is possible to prove that a multiple of 6 +- 1 has rest 1 in the division by 24.
x = (6k+-1)^2 mod 24
x = 36k^2 +- 12k + 1 mod 24
x = 12k^2 +- 12k +1 mod 24
x = 12 * k*(k +- 1) + 1 mod 24
And since k*(k +- 1)=0 mod 2, because it is the product of two consecutive integers (and therefore must be even)
x = 1 mod 24
Haha, yeah, that's basically what I did too and was wondering why he said it was too complicated... Started to think I did something wrong
Gustavo Exel are you German?
@@deept3215 Lol, I proved it too and was confused how you could make a 13 minute video on the properties without realizing it was trivial.
You missed a factor 3: x = 12*k*(3*k +/- 1) + 1 mod 24
I was so proud that my proof is the “simpler” proof. Although being in secondary school... maybe I had a headstart with the p^2 - 1 part.
I think the word for the second proof is "elegant", it's compact, gets the job done.
But elegance in design often comes after the working out and pruning of things that are unnecessary, and are often not the route that is taken by a pathfinder; instead, it's the shortest route that you can really only clearly see after you've made it to the destination. I always think of when I would be off-trail in the mountains and come across something interesting. The path I would take people on to come and view the interesting thing was usually much shorter than the route I took to discover it, because now I have the destination and you can find the "shortest route" to it. I think the mental path of discovery is very analogous, and I'm happy that Matt has made a point of showing the more circuitous paths, I think it really makes the journey seem more accessible to people and de-mystifies math and knowledge, which is all too often held up as unattainable and some sort of magic. Yea, once you point something out to other agents and experts in your space, people will start optimizing immediately, and the result of that peer-engagement usually has that sort of elegant and beautiful quality. But, often the most innovative ideas come from a mind that is just bent on finding "A" better way or "A" solution, and it's great to showcase that grit and brute-force and inelegance are not enemies of furthering understanding and knowledge, while at the same time, showing how engagement with other experts takes a "cool" idea, and turns it into something beautiful. --- Thanks Matt (If you're still reading comments on here 4 years later)
I recently found a marvelous pattern in the prime numbers! Every prime number is a prime number!
did you know all primes are indivisible by all numbers except itself and one?
... and conversely!
Aaaaand Matt Parker failed a square again. Typical.
Lots of Parker Square jokes, but your wording was the best lol
Thanks xD
-Typical- Classic.
As he would say, at least he gave it a try.
Horrendous!
"elegant"
... as in ...
"The friend's proof seems more elegant."
... might serve better, in context, than "easier".
Searched the comments to find this one. Elegant was the word he was searching for.
Great video. I've always been fascinated with primes. The first thing I did when I got my forst computer(a Commodore 64 (khz processor speed) was to write a prime number generator and then tweak it until it would run really fast. Gees, what a geek.
You might like Dave's Garage channel. He talks a lot about programming prime number finders as a kid on very early computers and optimizing the code and now he uses the same code to test the speeds between 100 different programming languages.
@@SkippiiKai Thanks, I'll check that out.
I was writing a program to check if a number is prime or not and I used this mathematical concept over there.
I just realized that though 2 and 3 don't fit into Matt's theory, but they can be applied to the concept in reverse manner, i.e,
(2*2 -1) and (3*3-1) divide 24 perfectly.
That helped in optimization of my solution.
thing about this dude is that hes real genuine. hes really skilled in what he teaches - because he enjoys it. hes real. and i appreciate that
Also known as the Parker 24
We should name everything he comes up with after him
I'd argue it's just another type of Parker square.
Actually the p^2= 24k-1 part works
It's the 6k+1 and 6k-1 being equal to p part that's worthy of being called 'Le Parker 6'
parker 139
2 & 3 aren't *real* primes?!! And I suppose hydrogen & helium aren't real elements? 😉
Only real elements are uranium and above.
Aaron P.. They are real primes, but different from all the others. There is no way a non prime number can be in between 1 and 2 or 1 and 3, so it's a bit obvious that 2 and 3 must be prime. 5 is the first prime that has a non prime between it and 1 (namely 4)
No. See, hydrogen and helium are the only real elements. Everything heavier are just metals *astronomy intensifies*
And gold isn't an element? As it's not a prime?
They're Parker primes. They fail to square to one more than a multiple of 24, but at least they gave it a go.
I'm a simple man. I see Parker and squares, I click like!
staffehn I remember when you still made videos
It's always cool to find other TH-camrs you (used to) watch in the comments.
Hey Matt, it's a way shorter to show that (6n+1)² or (6n-1)² are Multiples of 24 plus 1
For Example (6n+1)² = 36n²+12n+1 = 12( 3n²+n) +1
3n²+n is always a Even number
because if n is uneven you have 3*uneven²+uneven which alswes ends up beeing even because uneven+uneven = even
and if n is even you have 3*even²+even which is even, too
Therefore there is always a k from the natural numbers such that 3n²+n = 2k
With that you have 12( 3n²+n) +1 = 12*(2k)+1 = 24k+1
You can do the same with (6n-1)²
I prefer factoring to 12n(3n±1) + 1. For 12n(3n±1) to be a multiple of 24, you need n or (3n±1) to be even. If n is even, we're done. If n is odd, then 3n is odd, and adding or subtracting 1 gives an even number, so (3n±1) is even.
@@EnteiFire4 you can also use the p=6 plus or minus 1 fact, and note that of p-1 and p+1 in the factorization p^2-1=(p-1)(p+1), one is going to be a multiple of 6 and the other a multiple of 6 plus or minus 2 and so is a multiple of 4.
@@richardfredlund3802 I like that! Although I think the pair is either a multiple of 6 and a multiple of 4, *or* a multiple of 12 and a multiple of 2. That still works, though.
To put it another way, the product of any two consecutive even numbers is a multiple of 8. So the square of any odd number is one more than a multiple of 8. And since all primes past 2 are odd, all you need is that one of those factors is a multiple of 3.
Or more concisely: If 2 doesn't divide p, 8 divides p^2-1. If 3 doesn't divide p, 3 divides p^2-1. So if neither 2 nor 3 divide p, then 24 divides p^2-1.
2 and 3 work too. 2^2 is (24 * 1/8 + 1), and 3^2 is (24* 1/3 + 1). And since the multiplier is a fraction less than 1, I am with Matt on calling these two numbers as sub-prime.
I'm sure someone somewhere said this (and I haven't finished watching the video, so maybe they'll cover it?) but 2^2 - 1 = 3, and 3^2 - 1 = 8 .... and 3 x 8 is 24 :)
Just say “For primes 5 and greater”
I was thinking what could be added "if p^2 > 24, then...."
U mean all the primes?
2 and 3 are subprimes.
According to matt Parker
My teacher actually had me and his other students prove this on a test. He expected us to use equivalence classes in mod 24. The proof follows these steps:
1) Partition the set of all integers by all of the equivalence classes in mod 24.
2) Consider the classes as the range of numbers from -11 to 12 (these numbers are actually equivalence classes, so they represent the set of all integers).
3) Cross out all of the multiples of two and all of the multiples of three. (We’re left with the equivalence classes -11, -7, -5, -1, 1, 5, 7, and 11, all still in mod 24).
4) Square each number and minus one. The new numbers are 0, 24, 48, and 120, which are all multiples of 24.
Of course, this proof does not show that only primes have this property. It only shows that numbers which are not multiples of two or three have this property, and since all primes are not multiples of two or three, they have this property. So, there are definitely numbers that aren’t multiples of two or three but are not prime, just like Matt showed in the video (e.g. 25). Such numbers are those of which there are multiple prime factors and none of the prime factors are two or three. In the case of 25, its prime factorization is 5 and 5, so it is one of the numbers that is not a multiple of two or three and is not a prime number. But it is definitely true that prime numbers are not multiples of two or three, so they can be squared and end up being one more than a multiple of 24.
Why don't use mod30?
Then you are left with 8 possible primes every 30 numbers. 30 +-(1,7,11,13)
Just like in mod 24 bur you seive out more numbers.
I've been interested in and studies prime numbers since I was 14 years old, and next month I will be 74, so that's 60 years. I've found all sorts of interesting, quirky facts about them. They are some of the most fascinating numbers to study, because it seems like there should be no patterns and yet they are everywhere.
Using the same method as the second (more creative) proof, it also turns out that if you take the square of a prime number and multiply it by that same square minus five, you'll always end up with four less than a multiple of 360.
Example (using the prime number 7): 49 × 41 = 2156 = 2160 - 4, and 2160 = 360 × 6.
The proof comes from multiplying the factors (p - 2) (p - 1) (p + 1) and (p + 2). You'd end up with a polynomial that looks like p^4 - 5p^2 + 4, which can be rewritten as p^2 (p^2 - 5) + 4.
When you look at the four factors on a number line, in addition to having a multiple of 2, 3, and 4, the newly added (p - 2) and (p + 2) also guarantee a second multiple of 3 as well as a multiple of 5 (but only if you're using prime numbers higher than 5). Therefore, since 2 × 3 × 3 × 4 × 5 = 360, you can guarantee that multiplying all four factors will give you a multiple of 360.
I had no idea the primes could be divided into categories like this! In my (admittedly limited) maths education I got the impression that the defining characteristic is being absolutely without patterns. This video, as well as another video where you actually directly state that primes do have patterns, have enlightened me! Thank you. :)
Its not a generator, because not every (24k + 1 ) is prime. So its really not showing a pattern. Its created a pattern for possible primes, just the same as "not even" creates a pattern for possible primes. Now show a pattern to ALL the primes and ONLY the primes.
24 used to be my favorite number.
Many of the reasons why it was my favorite number is basically the same reason why some people suggest Dozenal is a better number system than Decimal, it just divides nicely by a lot of single digit numbers.
What is your current favorite number?
@@willmungas8964 Not sure I even have one any more. though I do like the powers of 2, like 16, 32 etc, and I do still like 24.
I like the second proof better not because it's "easier" but because it also shows why 2 and 3 don't square to multiples of 24 which is nice
This is just a really easy number theory problem. We just use the fact that all primes can be written as 6k+-1 excluding 2 and 3.
This fact is simple: We could have 6k,6k+1,6k+2,6k+3,6k+4,6k+5. Unless the prime is 2 or 3, we must have P=6k+5 or 6k+1.
6k+5 is the same as a number of the form 6k-1. So we're just squaring 6k+-1.
We just get P^2= 12(3k^2+-k)+1
3k^2-k is the same as (3k-1)k. Either k or 3k-1 will be even (If k is odd 3k-1 is even and if k is even k is even).
Then 3k^2+-k is the same as 2n for some integer n.
Plugging in gives us P^2=12(2n)+1=24n+1 with the exceptions of 2 and 3.
This isn't special about primes- Any number of the form (6k+-1)^2 is one more than a multiple of 24.
I'm definitely a Matt Parker type of maths enthusiast. I love maths, and I really appreciate the beauty of that second proof, but I would've for sure gone down the route of the first proof if I was solving this. I wish I had the intuition to solve problems the way the second proof does, but I don't.
Parker: Squaring Primes
Bruh mathematicians will pull some bogus like “this number has to either be equal to 1 or not equal to 1” and it somehow shows them the answer
Strange but true. Proof by cases can be very helpful. It’s also why most mathematicians do their best work while they are young and creative. The genius of many mathematicians comes from clever ways to rethink of problems in (relatively) simpler terms
Very interesting. I already knew that 17^2 is 289 because, well, I like numbers, especially primes, and just happened to know that. Incidentally, genius savant Daniel Tammet called 289 an "ugly" number (in his incredible synesthetic mind), but I find the number 289 quite lovely.
The most instructive thing about this video is Matt explaining the difference between doing a proof the "easy" way and doing it the "hard" way.
You can do it from the 6k+1 and 6k-1 cases.
Squaring 6k+1 gives 36kk + 12k + 1, which is 24(1.5kk + 0.5k) + 1.
Squaring 6k-1 gives 36kk - 12k + 1, which is 24(1.5kk - 0.5k) + 1.
If k is odd then k squared is odd, if k is even, then k squared is even - therefore the bit in brackets is an integer.
QED.
Hello
A simple proof can better be described as an elegant proof.
I had a geometry professor in community college always say, "Matthew, make this proof more elegant." At the time I didn't know what he meant. It wasn't until my capstone math course that I finally got what he meant. No other professor ever said it. I have my bachelors in math now. I'm with you. When he said easier, I immediately thought, nah that's more elegant.
@@Thedeadbeatmatt tbf, what is "easier" depends on where you're coming from at the moment. For me, the whole proposition seemed almost trivial and the p^2 - 1 approach sounded very similar to something I would try first. But that is because of something I have been working on that is actually very related to that, so of course I would try something more like it (that likely would quickly reduce to it itself).
"2 and 3, I call them the subprimes"
~Matt Parker "Square"
4:06 wow
Guy's secretly a cat.
Probably mentioned before, but I do like how 2^2-1=3 and 3^2-1=8, multiplying to form a familiar number.
Ooh! This is two years old and I have no idea what's in it but I love square primes
Hello everyone, for more codes number required send a message on my whatssap +1 972-534-5934
Nice haircut.
😂😂😂
Almost balding, not quite... could call it a parker cut.
The ears could still use a trim
@@kgipe
how would he look without ears ?
;)
Pleindespoir 🙉😂
The first is brute force, the second is elegant
Elegant proofs when clearly explained are usually more understandable. The brute force approach is arguably a stronger demonstration of primes occur next to 6. The elegant version requires the explanation to follow.
Want an even more elegant one?
All primes are +-1 mod 3, which means all prime number squares are 1 mod 3.
All primes are +-1 or +-3 mod 8 which means all prime number squares are 1 or 9 mod 8, and 9 is also 1 mod 8.
Combine those two facts to get that all prime number squares are 1 mod 24.
I think the "slight of hand" is in calling the subject primes when ANY number not a factor of 2 or 3 will fit that pattern.
My approach (which is very close to the (p+1)*(p-1) explanation:
1) Every prime number can be either expressed by 3a +1 or by 3a + 2. (a is an integer)
(3a+1)² = 9a² + 6a + 1 -> (3a+1)² - 1 can be divided by 3
(3a+2)² = 9a² + 12a + 4 = 9a² + 12a + 3 +1 -> (3a+2)² - 1 can be divided by 3
2) Every prime number can be expressed as 2b+1
(2b+1)² = 4b² + 4b + 1
-> if b=2c (i.e. b is even), then (2b+1)² = 16c² + 8c + 1 -> (2b+1)²-1 can be divided by 8 if b is even
-> if b=2c+1 (i.e. b is odd), then (2b+1)² = (4c+3)² = 16c² + 24c + 9 = 16c² + 24 + 8 + 1
-> (2b+1)²-1 can be divided by 8 if b is odd.
-> p²-1 can be divided by 8 and by 3 and therfore by 24...
Even easier: Ignoring 2 and 3, a prime is either 6m+1 or 6m-1. (6m+1)^2= 36m^2+12m+1 = 12m(3m+1)+1. If m is odd, m = 2k+1. 3(2k+1)+1=6k+3+1=6k+4=2(3k+2) which is divisible by 2 so you can factor out another 2 to get 24m(3k+2) + 1. If m is even, m=2k which means 12m=12(2k)=24k. So (6m+1)^2 is either 24m(3k+2)+1 or 24k(3m+1)+1. It works the same way if the prime is 6m-1.
Every fourth power of a prime except for 2, 3, and 5 is one more than a multiple of 240.
Every sixth power of a prime except for 2, 3, or 7 is one more than a multiple of 504.
Let k be integer and p(n) be the n-th prime number, then:
p(n>2)^2-1 = 1 x 24 x k
p(n>3)^4-1 = 10 x 24 x k
p(n>4)^6-1 = 21 x 24 x k
p(n>3)^8-1 = 20 x 24 x k
p(n>5)^10-1 = 11 x 24 x k
p(n>3)^12-1 = 2730 x 24 x k
p(n>2)^14-1 = 1 x 24 x k
p(n>7)^16-1 = 680 x 24 x k
p(n>8)^18-1 = 1197 x 24 x k
p(n>5)^20-1 = 550 x 24 x k
p(n>9)^22-1 = 23 x 24 x k
p(n>6)^24-1 = 5460 x 24 x k
...
As usual, the 24th power is a show off...
@@unfetteredparacosmian Mind=blown
5^6=15625=31*504+1
11^6=1771561=3515*504+1
13^6=4826809=9577*504+1
Every zeroth power of a prime is one more than a multiple of 8,200,601.
Hi Matt-- Thank you for this interesting episode. I really dig your presentation
Timestamps for the funniest parts
0:20
1:21
1:34
4:18
7:47
Whenever he was trying to compute 17^2 and was coming up with an easy way to do it, I immediately thought "that's gonna be 170 times 2, minus 3 lots of 17." I even paused the video and heard it in Matt's voice in my head. "170 times 2 is 340, 3 lots of 17, 51, 340 minus 51........ 289."
You can hear it in his voice now, can't you?
I have been doing something similar as a easy trick to multiply squared numbers in my head.
The difference of squares thing can be generalized.
So, a^2, can be modified to a^2 - s^2 and it be changed to (a-s)(a+s). To solve for a^2, just add the s^2 back on to the answer.
So, 19^2, can be rewritten as (19-1)(19+1) + 1 (or 361). 22^2 can be rewritten as (25)(19) + 9 or 484.
I did 17 squares in my head and got it right first try. I’m proud of myself.
Sounds easy enough. Just do it as (16+1)².
I am the 17th like of this comment. I am proud of myself.
Open question: I’m from Canada and when we talk about mathematics we shorten it to “math” not “maths” the way you do in UK, Aus, etc. Any reason why 4:28 said “Math-related items” vs “maths” despite Matt and Brady’s Aus backgrounds? Am I up too late again?
IKR they are not being consistent.
Still pretty sure, that delivery actually was a new role of wrapping paper to write on and a bunch of sharpies.
Placing 6n+-1 in several rows on a spreadsheet, 1 thru 6, 7 thru 12, 13 thru 18 etc. The result here is that two columns will contain prime numbers. Now 2 * 3 * 5 = 30. Place this on the spreadsheet 1 thru 30, 31 thru 60, 61 thru 90 etc. The result here is that 8 columns will contain prime numbers. Now 2 * 3 * 5 * 7 = 210. Place this on the spreadsheet 1 thru 210, 211 thru 420, 421 thru 630, etc. The result here is that 43 columns will contain prime numbers. If we continue with this 2310 gives 340 columns of primes, 30030 gives 3243 columns of primes. This continues with each prime number in the factorial. This also indicates the possible location of the different size gaps.
I think it is indeed worth emphasizing that the "prime" part of the statement is basically a red herring. And that's something I say as a number theorist! The claim is just about integers which aren't divisible by 2 or 3, i.e., numbers which are coprime to 24. In the world of abstract algebra, we would say that the (multiplicative) group (Z/24Z)* is isomorphic to the (additive) group (Z/2Z)x(Z/2Z)x(Z/2Z). In practical terms, that implies that if n is an integer which is coprime to 24, then n^2 is congruent to 1 mod 24.
Similarly, if you take any integer n which isn't divisible by 2, 3, 7 or 31, then n^(30) = 1 mod 5208. This is because:
(Z/5208Z)* = (Z/2Z)^5 x (Z/3Z) x (Z/5Z).
You can do similar things with 5208 replaced by any integer! You just need to look at the group structure. The only thing that makes some cases look special is finding an integer (like 24 in the video, or 5208 above) where the elements of the multiplicative group of units have small order compared to the size of the group. You can do this by finding a collection of primes p where p-1 is a "smooth" number.
This is some Grade-A prime content. I love prime facts.
0:50 parkerSquare(17) = 149.
Should we create a new OEIS sequence to collect all the parker squared Matt has discovered over the years?
Parker wrote 139, not 149.
I wanna know what was in that delivery was all about!!
It was a copy of the book "Arithmetic without Calculators"
Lol
He got a fresh shipment of math.
Pure, unrefined math.
3D printed buttplugs. With the profile specified by a complex formula.
Thank you for this very much for this video. The video shows that pi is approximately 22 / 7. This value is approximately 3.14. Using the properties of this value we can compute prime numbers in sequence, which is based on the existing computing capability. The formula was an algorithm, that was developed by a well known mathematician. Using his formula and the method that I discovered, I can compute prime numbers in sequence using 22 / 7 .
How do you do that?
how does this video relate to 22/7?
all prime number can be generated by fractran of 2 and fractran hides properties of the golden ratio
Easy way to find 17 squared: add 3 to get 20, subtract 3 to get 14, multiply 20 by 14 and add 3 squared.
(a+b)(a-b) = a*a - b*b, thus (a+b)(a-b) + b*b = a*a
Generally: choose smallest b so that a+b or a-b is a multiple of 10
0:17 i was told in school a bit ago that the were no patterns to prime numbers
they lied to you
they probably meant that theres no explicit formula to generate the sequence of primes (except something something mill's constant)
@@nathanisbored yeah i think thats what they meant
TECHNICALLY there is not any pattern. But when we talk about patterns, we mean patterns like the Fibonacci sequence, 1, 1, 2, 3, 5, 8... and so on. Primes doesn't have an algebraic formula to follow, but there are ways to figure out the nth prime number in other ways.
It really comes down to how you define a pattern though, which is a problem since everyone has unique definitions for just that.
You can set square root (24*n-1), n positive integer and create a prime sequence, removing irational numbers. It's a pattern, isn't it?
6:16 that looked like e (euler’s number) at first.
2 squared is 4
4-1 is 3
3 is 24×(1/8)
Not a *whole number* multiple of 24, but close.
3 squared is 9
9-1 is 8
8 is 24*(1/3)
Not a *whole number* multiple, but close.
Every prime that is not 3 gives a remainder 1 or 2 (mod 3). It is easy to show that the squares of numbers which give a remainder either 1 or 2 (mod 3) always give remainder 1 (mod 3). Also since every prime except 2 is an odd number, it is also easy to show that a square of any odd number always gives a remainder of 1 (mod 8). At the end, since 3 and 8 are relativley prime we can say that squares of prime numbers bigger than 3 always give remainder of 1 (mod 24) which is equivalent to saying that the squares are in the form 24k+1
The powers of prime numbers and the numbers that can only be divided by the different prime numbers are very common. For example: 2^x, 6^x, 30^x, 210^x, etc.