When the intuition is correct

Before starting college I read Paul Zeitz’s The Art and Craft of Problem Solving (a must read book along with the books by Arthur Engel and Terence Tao) and after the first example to illustrate Psychological Strategies he writes (pp. 15):

Just because a problem seems impossible does not mean that it is impossible. Never admit defeat after a cursory glance. Begin optimistically; assume that the problem can be solved. Only after several failed attempts should try to prove impossibility. If you cannot do so, then do not admit defeat. Go back to the problem later.

And today I will share a problem posed by August Ferdinand Möbius around 1840:

Problem of the Five Princes:
There was a king in India who had a large kingdom and five sons. In his last will, the king said that after his death the sons should divide the kingdom among themselves in such a way that the region belonging to each son should have a borderline (not just a point) in common with the remaining four regions. How should the kingdom be divided?

The hint is in the title of this blog post. The solution is easy, hence I won’t discuss it here. The reader is invited to write the solution as a comment to this post.

Prime Polynomial Theorem

I just wanted to point towards a nice theorem, analogous to the Prime Number Theorem, which is not talked about much:

# irreducible monic polynomials with coefficients in and of degree , for a prime power .

The proof of this theorem follows from Gauss’ formula:

# monic irreducible polynomialswith coefficients in and of degree = , by taking .

 

For details, see first section of this: http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf

Recursion and Periodicity

One of the simplest recursive formula that I can think of is the one which generates the Fibonacci sequence, with . So, I will illustrate a following general concept about recursions using Fibonacci sequence.

A sequence generated by a recursive formula is periodic modulo k, for any positive integer k greater than 1.

I find this fact very interesting because it means that a sequence which is strictly increasing when made bounded using the modulo operation (since it will allow only limited numbers as the output of recursion relation), will lead to a periodic cycle.

Following are the first 25 terms of the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025.

And here are few examples modulo k, for k=2,3,4,5,6,7,8

As you can see, the sequence repeats as soon as 1,0 appears. And from here actually one can see why there should be a periodicity.

For the sequence to repeat, what we need is a repetition of two consecutive values (i.e. the number of terms involved in the recursive formula) in the sequence of successive pairs. And for mod k, the choices are limited, namely k^2.  Now, all we have to ensure is that “1,0” should repeat. But since consecutive pairs can’ repeat (as per recursive formula) the repetition of “1,0” must occur within the first k^2.

For rigorous proofs and its relation to number theory, see: http://math.stanford.edu/~brianrl/notes/fibonacci.pdf

Under 40

The age of 40 is considered special in mathematics because it’s an ad-hoc criterion for deciding whether a mathematician is young or old. This idea has been well established by the under-40 rule for Fields Medal, based on Fields‘ desire that:

while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others

Though it must be noted that this criterion doesn’t claim that after 40 mathematicians are not productive (example: Yitang Zhang).  So I wanted to write a bit about the under 40 leading number theorists which I am aware of (in order of decreasing age):

• Sophie Morel: The area of mathematics in which Morel has already made contributions is called the Langlands program, initiated by Robert Langlands. Langlands brought together two fields, number theory and representation theory. Morel has made significant advances in building that bridge. “It’s an extremely exciting area of mathematics,” Gross says, “and it requires a vast background of knowledge because you have to know both subjects plus algebraic geometry.” [source]
• Melanie Wood: Profiled at age 17 as “The Girl Who Loved Math” by Discover magazine, Wood has a prodigious list of successes. The main focus of her research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, random groups, and algebraic topology.  [source1, source2]
• James Maynard:  James is primarily interested in classical number theory, in particular, the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly sieve methods, to study primes.  He has established the sensational result that the gap between two consecutive primes is no more than 600 infinitely often. [source1, source2]
• Peter Scholze: Scholze began doing research in the field of arithmetic geometry, which uses geometric tools to understand whole-number solutions to polynomial equations that involve only numbers, variables and exponents. Scholze’s key innovation — a class of fractal structures he calls perfectoid spaces — is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. Scholze’s work has a prescient quality, Weinstein said. “He can see the developments before they even begin.” [source]

Ulam Spiral

Some of you may know what Ulam’s spiral is (I am not describing what it is because the present Wikipedia entry is awesome, though I mentioned it earlier also). When I first read about it, I thought that it is just a coincidence and is a useless observation. But a few days ago while reading an article by Yuri Matiyasevich, I came to know about the importance of this observation. (Though just now I realised that Wikipedia article describes is clearly, so in this post I just want to re-write that idea.)

It’s an open problem in number theory to find a non-linear, non-constant polynomial which can take prime values infinitely many times. There are some conjectures about the conditions to be satisfied by such polynomials but very little progress has been made in this direction. This is a place where Ulam’s spiral raises some hope. In Ulam spiral, the prime numbers tend to create longish chain formations along the diagonals. And the numbers on some diagonals represent the values of some quadratic polynomial with integer coefficients.

Ulam spiral consists of the numbers between 1 and 400, in a square spiral. All the prime numbers are highlighted. ( Ulam Spiral by SplatBang)

Surprisingly, this pattern continues for large numbers. A point to be noted is that this pattern is a feature of spirals not necessarily begin with 1. For examples, the values of the polynomial form a diagonal pattern on a spiral beginning with 41.

Source: http://kvant.mccme.ru/1975/05/formuly_dlya_prostyh_chisel.htm