Goldbach conjecture
Every even whole number greater than two appears to be the sum of two primes. A proof covering all of them is still missing.
Prime patterns
Prime pairs, prime sums, and the distribution of primes through the lens of the zeta function.
Subjects: Number theory
Problems in this collection
7 problemsTwin prime conjecture
Prime pairs such as 11 and 13 are separated by two. Do such pairs keep appearing without end?
Riemann hypothesis
There are 25 primes up to 100 and 168 up to 1,000. Their uneven counts follow a broad trend; a conjecture about the zeta function predicts how far they can stray from a refined estimate.
Legendre’s conjecture
Must a prime number occur between n squared and the next integer’s square, however large n becomes?
Artin’s primitive root conjecture
For a fixed eligible integer, how often do its powers generate every nonzero residue modulo a prime?
Mersenne primes
Do prime numbers of the form 2 to a prime exponent, minus 1, continue forever?
Elliott–Halberstam conjecture
How far can averaged control of prime-counting errors in arithmetic progressions extend as the modulus grows?