every single odd number can be represented as the difference between two consecutive squares because the difference between two squares n and n+1 is (n+1)^2-n^2 = 2n+1. so 2n+1 = 101, 2n = 100, n = 50 and m = 50+1 =51. And then you can just flip through negatives to get the other solutions.
I admit, I like this better than my solution, because induction on n leads to the sum of sequential odd numbers (starting at 1) being a perfect square.
It's a good insight, but not a compete solution without factoring and observing that 101 is prime, to eliminate the possibility of other non-consecutive solutions.
(m - n)(m + n) = 101, a prime. So so one of these four pairs of equations is satisfied: m - n = 1 and m + n = 101 m - n = 101 and m + n = 1 m - n = -1 and m + n = -101 m + n = -1 and m - n = -101 There are two pairs of opposite-sign equations, meaning that m and n for those two are of opposite sign. Pick the top pair, and add the two equations to get 2m = 102 and m = 51. This makes n 50. The general case is m = ±51 and n = ±50. The signs need not match.
I will recommend 4 books: 1- Elementary Number Theory with Applications by Thomas Koshy (my fav) 2- Elementary Number Theory by James K. Strayer 3- Elementary Number Theory by Underwood Dudley 4- 250 Problems in Elementary Number Theory - Sierpinski (amazing collection of problems)
Isn't this a Hyperbola? I used Desmos If x^2-y^2 =101 => (10.05,0) & (-10.05,0) are its vertices and same applies for y^2-x^2=101 but the points will be on y axis. Any of this makes sense related to your provided solution?
Unless my memory is failing me, if a product of conjugates (a+b)(a-b) (which is what the difference of two squares translates to) equals a prime number, then their difference will be (a-b) = 1, and their sum (a+b) will equal the prime, and will be based on two numbers with a difference of 1. In this case, it's 51 and 50 without any brain work... Or 51^2 - 50^2 = 2601 - 2500 = 101. Perhaps you mention this in your video, but I didn't have 5 minutes to follow, sorry 🙂
every single odd number can be represented as the difference between two consecutive squares because the difference between two squares n and n+1 is (n+1)^2-n^2 = 2n+1. so 2n+1 = 101, 2n = 100, n = 50 and m = 50+1 =51. And then you can just flip through negatives to get the other solutions.
I admit, I like this better than my solution, because induction on n leads to the sum of sequential odd numbers (starting at 1) being a perfect square.
I wanted to say this!
It's a good insight, but not a compete solution without factoring and observing that 101 is prime, to eliminate the possibility of other non-consecutive solutions.
Easy peasy - thank goodness for prime numbers!
yess
(m - n)(m + n) = 101, a prime. So so one of these four pairs of equations is satisfied:
m - n = 1 and m + n = 101
m - n = 101 and m + n = 1
m - n = -1 and m + n = -101
m + n = -1 and m - n = -101
There are two pairs of opposite-sign equations, meaning that m and n for those two are of opposite sign. Pick the top pair, and add the two equations to get 2m = 102 and m = 51. This makes n 50. The general case is m = ±51 and n = ±50. The signs need not match.
Thanks Professor
You're welcome!
50;51 is the obvious
factor m^2-n^2 === (m+n)(m-n) & 101 is odd the trivial solution as above.
and singular as 101 is prime.
Recommend a good book for number theory please as u said in the video
I will recommend 4 books:
1- Elementary Number Theory with Applications by Thomas Koshy (my fav)
2- Elementary Number Theory by James K. Strayer
3- Elementary Number Theory by Underwood Dudley
4- 250 Problems in Elementary Number Theory - Sierpinski (amazing collection of problems)
This is a good book, shared by the author himself, on his webpage:
shell.cas.usf.edu/~wclark/elem_num_th_book.pdf
@@SyberMathf I may add, Elementary Number Theory by David Burton is also a fantastic book imo!
Isn't this a Hyperbola? I used Desmos
If x^2-y^2 =101 => (10.05,0) & (-10.05,0) are its vertices and same applies for y^2-x^2=101 but the points will be on y axis. Any of this makes sense related to your provided solution?
m = 10
n = ✓-1
m² - n² = 10² - (✓-1)²
100-(-1) = 101
If One Of Them Can Be Made Complex
n is not integer tho
Since m²-n²=101
It implies m²>n² 👍
Great work 👍
Thanks! 😃
m=51, n=50 (may be negative or positive)
Difference of squares= prime.
👍
Unless my memory is failing me, if a product of conjugates (a+b)(a-b) (which is what the difference of two squares translates to) equals a prime number, then their difference will be (a-b) = 1, and their sum (a+b) will equal the prime, and will be based on two numbers with a difference of 1. In this case, it's 51 and 50 without any brain work... Or 51^2 - 50^2 = 2601 - 2500 = 101. Perhaps you mention this in your video, but I didn't have 5 minutes to follow, sorry 🙂
Yeah' it is +-51 and +-50.
@@jonjuathanjoathan2426 Sure thing, negatives included, thanks for pointing that out!
i first solved in my brain
m = 11
n = √20
but then i saw "integers"???
and i was like 😭💔