World's Most Hardest Maths Problems

What is the Most Difficult Math Problem in the World?

There are two maths problems in the world that have received a lot of recognition and attention because they have remained unsolved for several years. While Riemann's Hypothesis still remains unsolved, Fermat's theorem which is one of the hardest math problems in the world, was solved only in 1995. Though difficult to understand, we will try and explain these two problems in the next section.

Riemann's Hypothesis
Put forward by Bernhard Riemann in 1859, the Riemann's Hypothesis is widely considered the most difficult math problem in the world. Riemann took forward the Euler's zeta function to all complex numbers barring s =1. On studying this further, Riemann realized that the zeta function had trivial zeros at -2, -4, -6, etc. and all non-trivial zeros were symmetric where the line Re(s) = ½. This led him to put forward the hypothesis that all non-trivial zeros are on the line Re(s) = ½. The Riemann hypothesis is stated as below.

The real part of any non-trivial zero of the Riemann zeta function is ½, and thus, the non-trivial zeros should lie on the critical line, ½ + it where i is an imaginary unit and it is a real number.

Fermat's Theorem
Fermat's theorem or Fermat's Last Theorem as it is known, was put forward by Pierre de Fermat in 1637. After several years of many mathematicians, trying to prove the theorem, it was solved after more than three hundred years in 1995. Fermat's theorem is stated as below.

No three positive integers a, b, and c can prove the equation an + bn = cn, for any integer n, whose value is greater than 2.

While this theorem was proved for the integer case n=4 before Fermat's theorem was proposed, over the next two hundred years, the theorem was proven for the prime numbers 3, 5, and 7. The theorem was over the years proved for all prime numbers less than 100 and for regular primes. It was in 1984 that Gerhard Frey proposed that the theorem could be proved using the modularity conjecture. Andrew Wiles successfully proved the Fermat's Last Theorem in 1995, with the assistance of Richard Taylor.

Fermat's Last Theorem was published only after his death, as when he was alive, Fermat, an amateur mathematician refused to publish any of his work. In fact, the theorem was scrawled on the margins of one of his books and found later by his son. Along with the yet unproven Riemann's hypothesis, Fermat's last theorem is without doubt the hardest math problem in the world.

Both these theorems have achieved cult popularity in mathematical circles, seeping into popular culture with mentions in bestselling books like the Millennium Trilogy by Steig Larrson and series like Simpsons, Numb3rs, and Law and Order. So what if mere mortals like us cannot harbor any hopes of solving the hardest mathematics problem in the world, we can at least look intelligent while mentions are made. Like Bertrand Russel once said, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."