Although I am already a number theory junkie, my beast of choice being the 3x3 magic square of integer squares (maybe my name gives that away), your enthusiasm is truly infectious. Many thanks for sharing this lecture.
The field is principally devoted to consideration of direct problems over (typically) the integers, that is, determining the structure of hA from the structure of A: for example, determining which elements can be represented as a sum from hA, where A is a fixed subset.[1] Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2P contains all even numbers greater than two, where P is the set of primes) and Waring's problem (which asks how large must h be to guarantee that hAk contains all positive integers, where A k = { 0 k , 1 k , 2 k , 3 k , … } {\displaystyle A_{k}=\{0^{k},1^{k},2^{k},3^{k},\ldots \}} A_k=\{0^k,1^k,2^k,3^k,\ldots\} is the set of k-th powers). Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes. Hilbert proved that, for every integer k > 1, every nonnegative integer is the sum of a bounded number of k-th powers. In general, a set A of nonnegative integers is called a basis of order h if hA contains all positive integers, and it is called an asymptotic basis if hA contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set A is called a minimal asymptotic basis of order h if A is an asymptotic basis of order h but no proper subset of A is an asymptotic basis of order h. It has been proved that minimal asymptotic bases of order h exist for all h, and that there also exist asymptotic bases of order h that contain no minimal asymptotic bases of order h. Another question to be considered is how small can the number of representations of n as a sum of h elements in an asymptotic basis can be. This is the content of the Erdős-Turán conjecture on additive bases. en.wikipedia.org/wiki/Additive_number_theory
Simon Singh, 'Fermat's Last Theorem' prime page 47 says ' Fermat's greatest love was for a subject which is largely useless - the theory of numbers.' Launcelot Hogben, 'Mathematics For The Millions' said words to the effect that ' when clever people lose contact with the common man they are in danger of becoming a priesthood. I agree with them both! Welcome to 'The Holy Tabernacle of Useless Mathematics' where within resides 'The Holy Arc of The Covenant of Abstract Mathematics'.
There's a lot of research now in number theory, certainly important in computing as we want to ensure that our program won't crash if we put in some random entry, e.g. y = 1/(n-m) will more than likely work always, but we must ensure that there will never occur 1/0 result. We need mathematical tests to ensure that input x -> output y always. Another problem is the amount of computational power required to solve a particular problem. Type pi into your calculator and you should know that the calculator runs a sequence to generate the value. These are all real life issues in computing. My assertion is that Number Theory is not as abstract but is very much applied. You might want to think about encryption and its relation to known large prime numbers, that could be classified as literally a life or death problem!
@@tomctutor As I am still fortunate enough to be still on the planet, I must thank you for reminding me of this comment I made 9 years ago. I liked it then and I like it even more now. I didn't make it because I don't like maths, I do, i just don't like the so called 'New Mathematics' which leaves me cold and does little or nothing to directly improve my knowledge and understanding. It's indirect benefit to me is that by being antagonistic to it I have discovered for myself arithmetical, algebraic and geometric truths that do mean something to me and enhance my knowledge and understanding even though it is not to the liking of the establishment who label persons like me cranks, which for me it is an endorsement of what I do. I see that you have maths videos on your channel which look interesting so I will take the time to look at some.
Noddy's Thm: Every whole number N is the sum at most N-smaller numbers! No wait, every N(>4) is the sum of at least two smaller numbers, that's a lot better. -> Goldbach's conjecture, both of these numbers can be prime. Hmm, very satisfying but not so obvious.
Dr Vicky Neale should be congratulated for her summarizing many theorems in the past for all audiences who attended her lecture sponsored by the Institute of Education in London. I would like to take this opportunity to share with all audiences by sending to you a very interesting problem relevant to the number theory to watch by clicking the following website: plus.google.com/118141546473679947918/posts/6LWrsti19n1
prime numbers are numbers that cannot be factored by any other prime than itself by definition. if 1 was prime then no number other than 1 would be prime that's why 1 is not prime. If 1 was prime (say) 13=13.1 which is obviously a prime wouldn't be a prime because it would be a product of ''two different'' primes.
I don't know whether i am attending a lecture on number theory or visiting a circus ! Mr. Perelman must have seen her "performing" to have said "...the animal on zoo.." comment !
The first 30 minutes is WAFFLE But yes it is okay But not as amazing as I thought it would be :D I liked it hough any suggestions on what I should watch next?
With squares, I think 8 columns are better because quadratic residues mod 8 (remainders of the division of a square by 8) are just 0, 1 and 4; while mod 32 there are 7 quadratic residues (0,1,4,9,16,17,25) so with 8 columns it should be easier to find patterns. Is there a reason why you suggested using 32? I'm just curious, maybe I'm completely wrong...
Anyone who could put on a pants like that can’t be all that bad. 😉 By definition, there can be no negative primes. The very fact that there is no prime number symmetry extended to the negative axis probably implies that the whole notion of primes is a baseless man-made pipe dream, with no underlying significance. And they are always looking for the largest prime number and checking to see if they would ever run out or not, and yet they could hardly extend the search to the transfinite realms! Yes, we’ve found some curious patterns and interesting observations regarding prime numbers, but so what? You look hard enough, you could find interesting patterns in just about anything, some even with corresponding observations in nature (the Fibonacci numbers etc.). The reveries of number theorists has always been highly entertaining.
She is very fast where she makes the point to think..Only i hear blah blah multiple of eight.. blah blah multiple of eight. blah blah multiple of eight. blah blah multiple of eight....ANd i rewind what point is she making...Its really hard to understand who goes quick where a little time is required to atleast understand verbally ,what she is saying
A great mathematician, but a even greater educator!! Respect!!
unfortunately deceased, just watched her here for the first time and found out, made me so sad
Her way of explaining is so amazing
Although I am already a number theory junkie, my beast of choice being the 3x3 magic square of integer squares (maybe my name gives that away), your enthusiasm is truly infectious. Many thanks for sharing this lecture.
Great lectures, this lecture is the mandatory cross roads for future advanced research in the fields of applied quantum physics at the Cern.
Really good lecture here.
Hardy & Wright is one of my favourite books. I didn't know there was a new edition. Just ordered it. I can't wait.
Is it any good?
@@ryacoli Slightly disappointing. Mainly a new chapter on elliptic functions probably covered better elsewhere these days.
The field is principally devoted to consideration of direct problems over (typically) the integers, that is, determining the structure of hA from the structure of A: for example, determining which elements can be represented as a sum from hA, where A is a fixed subset.[1] Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2P contains all even numbers greater than two, where P is the set of primes) and Waring's problem (which asks how large must h be to guarantee that hAk contains all positive integers, where
A k = { 0 k , 1 k , 2 k , 3 k , … } {\displaystyle A_{k}=\{0^{k},1^{k},2^{k},3^{k},\ldots \}} A_k=\{0^k,1^k,2^k,3^k,\ldots\}
is the set of k-th powers). Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes. Hilbert proved that, for every integer k > 1, every nonnegative integer is the sum of a bounded number of k-th powers. In general, a set A of nonnegative integers is called a basis of order h if hA contains all positive integers, and it is called an asymptotic basis if hA contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set A is called a minimal asymptotic basis of order h if A is an asymptotic basis of order h but no proper subset of A is an asymptotic basis of order h. It has been proved that minimal asymptotic bases of order h exist for all h, and that there also exist asymptotic bases of order h that contain no minimal asymptotic bases of order h. Another question to be considered is how small can the number of representations of n as a sum of h elements in an asymptotic basis can be. This is the content of the Erdős-Turán conjecture on additive bases. en.wikipedia.org/wiki/Additive_number_theory
46:40 ...On the twin prime conjecture.... Any of you sorted that excercise out yet? LOL very fine sense of humor!
Simon Singh, 'Fermat's Last Theorem' prime page 47 says ' Fermat's greatest love was for a subject which is largely useless - the theory of numbers.' Launcelot Hogben, 'Mathematics For The Millions' said words to the effect that ' when clever people lose contact with the common man they are in danger of becoming a priesthood.
I agree with them both!
Welcome to 'The Holy Tabernacle of Useless Mathematics' where within resides 'The Holy Arc of The Covenant of Abstract Mathematics'.
There's a lot of research now in number theory, certainly important in computing as we want to ensure that our program won't crash if we put in some random entry, e.g. y = 1/(n-m) will more than likely work always, but we must ensure that there will never occur 1/0 result. We need mathematical tests to ensure that input x -> output y always.
Another problem is the amount of computational power required to solve a particular problem. Type pi into your calculator
and you should know that the calculator runs a sequence to generate the value. These are all real life issues in computing. My assertion is that Number Theory is not as abstract but is very much applied. You might want to think about encryption and its relation to known large prime numbers, that could be classified as literally a life or death problem!
@@tomctutor As I am still fortunate enough to be still on the planet, I must thank you for reminding me of this comment I made 9 years ago.
I liked it then and I like it even more now.
I didn't make it because I don't like maths, I do, i just don't like the so called 'New Mathematics' which leaves me cold and does little or nothing to directly improve my knowledge and understanding.
It's indirect benefit to me is that by being antagonistic to it I have discovered for myself arithmetical, algebraic and geometric truths that do mean something to me and enhance my knowledge and understanding even though it is not to the liking of the establishment who label persons like me cranks, which for me it is an endorsement of what I do.
I see that you have maths videos on your channel which look interesting so I will take the time to look at some.
I have no idea what’s going on but her accent is awesome
breathtaking
Noddy's Thm: Every whole number N is the sum at most N-smaller numbers! No wait, every N(>4) is the sum of at least two smaller numbers, that's a lot better. -> Goldbach's conjecture, both of these numbers can be prime. Hmm, very satisfying but not so obvious.
thank you
between four and seven is one prime number
Dr Vicky Neale should be congratulated for her summarizing many theorems in the past for all audiences who attended her lecture sponsored by the Institute of Education in London.
I would like to take this opportunity to share with all audiences by sending to you a very interesting problem relevant to the number theory to watch by clicking the following website:
plus.google.com/118141546473679947918/posts/6LWrsti19n1
Sorry Vicky, I can't let it go. Why is 1 not a prime number?
because we said so. Maths is generated from some arbitrary made up axioms.
prime numbers are numbers that cannot be factored by any other prime than itself by definition. if 1 was prime then no number other than 1 would be prime that's why 1 is not prime. If 1 was prime (say) 13=13.1 which is obviously a prime wouldn't be a prime because it would be a product of ''two different'' primes.
@@titanarmy4116 not all of mathematics is the product of made up axioms
I don't know whether i am attending a lecture on number theory or visiting a circus ! Mr. Perelman must have seen her "performing" to have said "...the animal on zoo.." comment !
The first 30 minutes is WAFFLE
But yes it is okay
But not as amazing as I thought it would be
:D I liked it hough any suggestions on what I should watch next?
Would have made more sense to use 32 columns
With squares, I think 8 columns are better because quadratic residues mod 8 (remainders of the division of a square by 8) are just 0, 1 and 4; while mod 32 there are 7 quadratic residues (0,1,4,9,16,17,25) so with 8 columns it should be easier to find patterns. Is there a reason why you suggested using 32? I'm just curious, maybe I'm completely wrong...
👌👍👍
I think you’re a prime number
The rushy train
Anyone who could put on a pants like that can’t be all that bad. 😉
By definition, there can be no negative primes. The very fact that there is no prime number symmetry extended to the negative axis probably implies that the whole notion of primes is a baseless man-made pipe dream, with no underlying significance. And they are always looking for the largest prime number and checking to see if they would ever run out or not, and yet they could hardly extend the search to the transfinite realms! Yes, we’ve found some curious patterns and interesting observations regarding prime numbers, but so what? You look hard enough, you could find interesting patterns in just about anything, some even with corresponding observations in nature (the Fibonacci numbers etc.). The reveries of number theorists has always been highly entertaining.
She said "particularly positive whole numbers" so are there any negative whole numbers?? Lol 😂😌
Whole numbers are numbers without a fraction or decimal place, AKA integers. And integers can be negative.
Some members of the audience might not know the distinction between whole numbers and integers, the layman!
She is very fast where she makes the point to think..Only i hear blah blah multiple of eight.. blah blah multiple of eight. blah blah multiple of eight. blah blah multiple of eight....ANd i rewind what point is she making...Its really hard to understand who goes quick where a little time is required to atleast understand verbally ,what she is saying
It seems she is the only 1 understanding whatever she is blabbering!
That's because you are a not PhD in maths. You know trust the doc when they say you need a vaccine.
Stop barking