Last week, I tackled two famous unsolved math problems: 1. Twin primes conjecture and 2. Goldbach conjecture. Today I discuss two more, very different. The first is almost a joke (extremely challenging!). The second is considered the most important of all, for its numerous consequences.
3. The Collatz Conjecture is quite easy to explain. Consider a positive integer N. If N is even, substitute N/2; if odd, replace with 3xN+1. Repeat successively. For example, N=14 is replaced by 14/2=7, which is replaced by 3×7+1=22, which is replaced by 22/2=11, and so on.
The German Lothar Colatz proposed this procedure in 1937, stating that it always ends up at number 1, whatever the initial N. The simplicity of the statement is deceiving. Hungarian expert Paul Erdös warned that “mathematics may not be ready for these kinds of problems”. And he offered 500 dollars for the solution.
Computationally, we know that it holds for all numbers up to twenty digits. In 1976, Riho Terras proved that for “almost all” N the sequence ends up taking values ​​lower than the initial N. This was improved upon by Terence Tao in 2019. It is encouraging, but to prove the conjecture will require new ideas.
4. Riemann’s hypothesis. In 1859, Bernhard Riemann wrote a certain formula ζ(x), called the zeta function. It had already appeared in Euler’s work from 1740, but Riemann extended the definition to the complex numbers x, and showed that this function tells us a lot about prime numbers.
A crucial question was what the zeros are, that is, the values ​​of x such that ζ(x)=0. Aside from the negative pairs -2, -4, -6, etc., Riemann knew that there are many other zeros, and he believed that they all have a real part equal to 1/2. Not being able to prove it, he accepted this fact as a hypothesis, deducing various results from it. Many mathematicians have since done the same, resulting in dozens of “provisional” theorems, whose validity depends on someone proving the hypothesis.
Hence, this problem appears on every list of mathematical problems, from Hilbert’s famous list at the International Congress of Mathematicians in 1900, to the 7 Millennium Problems, distinguished by the Clay Institute with prizes of $1 million. Hilbert said, “If I woke up after sleeping for a thousand years, my first question would be: Has the Riemann Hypothesis been proved?”
