## 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