13 problems

Let $F_n=2^{2^n}+1$ for $n\geq 0$ be the Fermat numbers. Fermat's conjecture. All Fermat numbers $F_n$ are prime. This conjecture was disproved by Euler in 1732, who proved that…

For an integer $n>1$, define … Giuga's conjecture. The integer $n$ is prime if and only if … This is a power-sum characterization of primality, presented in the source as Giuga's c…

For an integer $p>1$, let $\mathbf{B}_{p-1}$ denote the $(p-1)$-st Bernoulli number. Agoh–Giuga conjecture. The integer $p$ is an odd prime if and only if … This is a Bernoulli-num…

Let $E$ be a fixed elliptic curve over $\mathbb{Q}$, let $P\in E(\mathbb{Q})$ be a rational point, and let $m$ be the fixed integer indexing the generalized elliptic Fermat numbers…

Let $A_n=n!(n-1)!/2^{n-1}$, so that the sequence begins $1,1,3,18,180,2700,56700,1587600,57153600,\dots$. Schulte's conjecture. For all integers $n>1$, $n$ divides $2A_{n-1}+4$ if…

Let $E$ be a fixed elliptic curve over $\mathbb{Q}$, let $P\in E(\mathbb{Q})$ be a rational point, and let $F=(F_n)_{n\in\mathbb{N}}$ be the associated sequence of elliptic Fermat…

Let $D=(D_n)_{n\in\mathbb{N}}$ be an elliptic divisibility sequence. Primality conjecture. The sequence $D$ contains only finitely many prime terms. Everest, Miller, and Stephens p…

Wu's divisibility-by-3 conjecture. If $c(n)=0$, then $n$ is divisible by $3$.

OEIS A256480 conjecture.

Wu's divisibility-by-3 conjecture. If $n$ is not a multiple of $11$ with an even number of digits and $a(n)=0$, then $n$ is a multiple of $3$.

OEIS A090287 conjecture. The number $f(d,m,n)$ is composite for all integers $m>0$ and $0<d<10$ if and only if $n$ is a palindrome with an even number of digits; equivalently,…

Let $m$ be a positive integer. The binomial-sum primality conjecture. If $m\neq30$ and … then $4m-1$ is prime. If … then $4m+1$ is prime. The authors report checking $m\leq1500$ wi…

Let $B=(B_n)$ be an elliptic divisibility sequence generated by a rational point on an elliptic curve in minimal form. A prime term is a term $B_n/B_1$ that is prime. The uniform p…