# prime

## Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

\begin{conjecture} Every even number greater than $10$ can be represented as the sum of an odd prime number and an odd \Def {semiprime} . \end{conjecture}

## Polignac's Conjecture ★★★

Author(s): de Polignac

\begin{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. \end{conjecture}

In particular, this implies:

\begin{conjecture} Twin Prime Conjecture: There are an infinite number of twin primes. \end{conjecture}

## Are there an infinite number of lucky primes? ★

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

\begin{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. \end{conjecture}

## Twin prime conjecture ★★★★

Author(s):

\begin{conjecture} There exist infinitely many positive integers $n$ so that both $n$ and $n+2$ are prime. % You may change "conjecture" to "question" or "problem" if you prefer. \end{conjecture}

Keywords: prime; twin prime

## Wall-Sun-Sun primes and Fibonacci divisibility ★★

Author(s):

\begin{conjecture} For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once. \end{conjecture}

Equivalently:

\begin{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. \end{conjecture}

## Goldbach conjecture ★★★★

Author(s): Goldbach

\begin{conjecture} Every even integer greater than 2 is the sum of two primes. \end{conjecture}

Keywords: additive basis; prime