Every now and then, readers ask me to write about the most important unsolved problems in mathematics. Some are too complicated for a newspaper, they would take up too much space just to explain what it is about. But there are several that you can comment on here, and these are really the most interesting. I begin the list with two famous number theory problems.
1. Twin primes conjecture. This is the oldest unsolved problem, dating back to ancient Greece. Twin primes are pairs of prime numbers whose difference is equal to 2, for example 41 and 43.
As we consider larger numbers, the primes become more and more widely spaced (this is explained by the Prime Number Theorem, which I will talk about here on another occasion). The same goes for twin primes, only much faster: we know that twin primes are much rarer than primes. The question is to prove that, even so, the number of twin primes is still infinite.
In 2013, Yitang Zhang proved that there is an N number and an infinite number of prime pairs whose difference is at most N. Initially, N was huge (70 million!), but an international network of mathematicians led by Terence Tao narrowed it down. for N=246. Arriving at N=2 would prove the conjecture, but new ideas will be needed for that.
2. Goldbach’s conjecture. In a letter sent to Leonhard Euler on June 7, 1742, the German Christiab Goldbach proposed the following conjecture: every integer greater than 5 can be written as the sum of three primes (for example, 33=23+7+3). Euler responded on 30 of the same month, noting that this would be the same as showing that every even integer greater than 2 is the sum of two primes (for example, 42=23+19). And he added: “I consider this a guaranteed theorem, although I am not able to prove it.”
We still don’t know how to prove it, although the statement has been checked computationally for all numbers up to 18 digits. Furthermore, in 2013, the Peruvian mathematician of German origin Harald Helfgott proved that every odd integer greater than 7 is the sum of three primes. This is called the weak Goldbach conjecture because it is not enough to prove the original conjecture, but it is a good indication of its truth: if the original conjecture was false, the weak conjecture would have to be too.
