Unsolved Problems in Mathematics

This is an explanation of unsolved problems in mathematics. Although the problems themselves are simple, there are still many that have not been proven.

Are there infinitely many perfect numbers?

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For example,

• 6 has divisors 1, 2, and 3, and 1+2+3=6, so it is a perfect number.
• 28 has divisors 1, 2, 4, 7, and 14, and 1+2+4+7+14=28, so it is a perfect number.

Currently, only 51 perfect numbers have been discovered. It is conjectured that there are infinitely many, but this has not yet been proven.

Goldbach’s Conjecture

Goldbach’s conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. (Here, a prime number is a natural number that has no positive divisors other than 1 and itself.)

For example,

• 4=2+2
• 6=3+3
• 8=3+5
• 10=3+7=5+5

The problem itself is very simple, but it remains unproven.

Riemann Hypothesis

The Riemann hypothesis is the conjecture that the zeros of the Riemann zeta function are restricted to negative even integers and complex numbers with real part 1 / 2.

The Riemann zeta function is the function $\zeta$ defined by:

where $s$ is a complex number and $n$ is a natural number.