## Testing conjectures on primes

Goal:

Prime numbers are very mysterious objects and they have always fascinated people. Many basic questions about prime numbers are still open, for instance:

• (Goldbach conjecture) Is every even positive integer the sum of two primes?
• (Twim prime conjecture) Are there infinitely many prime numbers p such that p+2 is also prime?
• (Riemann conjecture, equivalent formulation) Let Pi(x) be the number of primes less than x. Is the absolute value of the difference Pi(x) - x/log(x) bounded by the square root of x divided by log(x) for all large enough x?

The goal is to test some conjectures on prime numbers on a computer. Possible conjectures to test are next to the many famous conjectures (including those mentioned above) also those by the Chinese mathematician Zhi-Wei Sun who is currently making an enormous number of new ones (see for instance this article).

This subject is related to the project on the distribution of primes.

Participants: Joel Costa da Rocha, Alex Ferreira Costa.

Supervisors: Gabor Wiese, Panagiotis Tsaknias.

Difficulty level: Experimental Mathematics 1.

Tools: Computer Algebra system like SAGE.

Results: Project report