Last week I wrote an article outlining a few simple yet unsolved problems in mathematics. At the time I had thought it would be a one off posting, rather than the start of a series of articles. However based on the positive responses I have received and the numerous inspiring mathematical discussions it has provoked, it is clear that there is a demand in society for more of these type of mathematical articles.

And so here we go again...

The last article that I wrote focused on colourings and on graph theory, and summarized three problems that were simple to understand and yet remain unsolved. Today I will shift to another field of mathematics, number theory. (And before anyone gets scared off, that just means these problems focus on the properties of the counting numbers). It should also be noted that although they are among the most famous unsolved problems in number theory, I am deliberately excluding both the Riemann hypothesis and the abc-conjecture from this article, as I have written about them several times in the past.

The Erdos-Strauss Conjecture:  There is an interesting property of the integers (or counting numbers) in that for every integer n, the number four can be written as the sum of exactly three fractions, each of which has the numerator n. In other words, for any integer n, you can choose three integers x,y,z such that

n/x + n/y + n/z = 4

At present this property has been tested for an unimaginably large number of integers, and it is always true. And yet no one has ever been able to prove that it should always be true. It is quite possible that there exists some enormous number for which it is not true. And yet it seems so simple...
Pillai's Conjecture:  This one is a little more complicated, but still simple enough to provide a fun challenge for amateur mathematicians. Select three integers, A,B,C, and then consider the equation
A xn + B ym = C

where m,n,x,y are all unknown integers. Pillai's conjecture is that for any choice of A,B,C, there will be only a finite number of solutions of this equation. 

Grimm's Conjecture:  As most of you will recall from primary school mathematics, a composite number is one that can be written as a product of integers other than itself and one. For example 12 can be written as 3*4, but 13 can only be written as 1*13. Grimm's conjecture states that for every set of k consecutive composite numbers, (n, n+1, n+2,n+3 ... n+k), there exists a set of k different prime numbers (p1,p2,...pk) such that pi divides n+i. For example the set of composite numbers (14,15,16) has the set of prime numbers (7,5,2). However it is unknown whether there are any sequences of composite numbers for which no set of distinct prime divisors exists.

As with the previous three problems that I wrote about in the first article, each of these three is (somewhat) simple to state, and yet in each case the greatest mathematical minds have been unable to prove them. I hope you enjoy them, and spend many entertaining hours pondering the beauty of number theory!