Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

Conjecture   Every even number greater than $ 10 $ can be represented as the sum of an odd prime number and an odd semiprime .

Keywords: prime; semiprime

Polignac's Conjecture ★★★

Author(s): de Polignac

Conjecture   Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

In particular, this implies:

Conjecture   Twin Prime Conjecture: There are an infinite number of twin primes.

Keywords: prime; prime gap

Are there an infinite number of lucky primes?

Author(s): Lazarus: Gardiner: Metropolis; Ulam

Conjecture   If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.

Keywords: lucky; prime; seive

Twin prime conjecture ★★★★


Conjecture   There exist infinitely many positive integers $ n $ so that both $ n $ and $ n+2 $ are prime.

Keywords: prime; twin prime

Wall-Sun-Sun primes and Fibonacci divisibility ★★


Conjecture   For any prime $ p $, there exists a Fibonacci number divisible by $ p $ exactly once.


Conjecture   For any prime $ p>5 $, $ p^2 $ does not divide $ F_{p-\left(\frac p5\right)} $ where $ \left(\frac mn\right) $ is the Legendre symbol.

Keywords: Fibonacci; prime

Goldbach conjecture ★★★★

Author(s): Goldbach

Conjecture   Every even integer greater than 2 is the sum of two primes.

Keywords: additive basis; prime

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