Some open problems involving Prime numbers

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Prime numbers are the most fascinating to mathematicians. Understandably so, as they are the basic atoms which uniquely constitute each number (Fundamental Theorem of Arithmetic) and also have very important applications in cryptography which is largely responsible for our internet security.  Over the centuries, we have learned a lot about them, yet there is an abundance of problems we still cannot prove. In this article, we will highlight a few of them.

Goldbach’s conjecture



Every even integer greater than 2 can be expressed as the sum of two prime numbers.

E.g. 34=3+31

But is it always true?

Legendre’s conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and (n+1)2 for every positive integer n. For example, between 4 and 9 which are square of 2 and 3, we have 5 and 7 as prime numbers.  The conjecture is one of Landau’s problems (1912) on prime numbers; as of 2020, the conjecture has neither been proved nor disproved.

Twin primes conjecture states that here are infinitely prime numbers that differ by just 2.

For example, 3 and 5, 11 and 13,137 and 139.

But, so far, there has been no proof that it holds in all cases.

The other aspect of prime numbers that fascinates mathematicians is the occurrence and distribution of prime numbers. Mathematicians believe that despite the apparent randomness of prime numbers, there is some undiscovered higher order of rules which govern them.  Initially, they occur frequently but as we go further along they become sparser. Below we show the first 1,000 prime numbers


 In fact, many attempts have been made to generate prime numbers. The following was suggested by Euler in the 18th century. 

Euler’s prime–generating polynomial

n=1 gives 41 (prime)
n=2, gives 43 (prime)
n=3, gives 47 (prime), etc.
Fails for the first time when n=40.

Among the most celebrated results in number theory is the Prime number theorem. The number of primes less than any given number N is roughly


The estimate gets closer and closer to the actual numbers of primes for larger and larger values of N. This does not tell us where the next prime number lies but does tell us how many primes are in a given number interval provided the interval is large enough.

However, the most publicized and important unsolved problem in all of mathematics is the Riemann hypothesis.


§  concerns the distribution of prime numbers

Please read my blog article on Riemann hypothesis for more details.

In conclusion, we still have a long way to go or if ever to truly understand prime numbers. Some of the unproven problems are tantalizing easy to state and understand yet notoriously difficult to prove. If you are willing to put in the work to prove any one of them and are able to do so successfully, I assure you of instant mathematical stardom and celebrity!

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