Unsolved Problems

Conjecture Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product....

Conjecture Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \bmod {b_i}}$ for $i \geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = ...

Definition A set of m positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$-tuple if $a_i\cdot a_j + 1$ is a perfect square for all...

Conjecture Let $p$ be a prime natural number. Find all primes $q\equiv1\left(\mathrm{mod}\: p\right)$, such that $2^{\frac{\left(q-1\right)}{p}}\equiv...

Conjecture Find linear transformation construction of primitive pythagorean n-tuple tree!...

Conjecture $3~$ is a primitive root modulo $~p$ for all primes $~p=16\cdot q^4+1$, where $~q>3$ is prime....

Conjecture For all $n > 2$, there exist positive integers $x$, $y$, $z$ such that $$1/x + 1/y + 1/z = 4/n$$....

Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the fol...

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

Conjecture $p$ is a prime iff $~\displaystyle \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p$...

Definition Let $r_i$ be the unique integer (with respect to a fixed $p\in\mathbb{N}$ ) such that $$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0...

Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer et...

Problem Let the notation $a|b$ denote " $a$ divides $b$ ". The mimic function in number theory is defined as follows [1]. Definition For any positiv...

Conjecture Skewes' number $e^{e^{e^{79}}}$ is not an integer....

Conjecture If $a_1,a_2,\ldots,a_{2n-1}$ is a sequence of elements from a multiplicative group of order $n$, then there exist $1 \le j_1 < j_2 \ldots <...

Conjecture For every $0 \le t \le n-1$, the sequence in ${\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewes...

For every finite multiplicative group $G$, let $s(G)$ ( $s'(G)$ ) denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a ...

Problem Does for every integer $N$ exist a covering system with all moduli distinct and at least equal to~ $N$?...

Problem (1) Prove that there exist infinitely many positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$ Problem (2) Prove th...

Problem Find an explicit formula for Frobenius number $g(a_1, a_2, \dots, a_n)$ of co-prime positive integers $a_1, a_2, \dots, a_n$ for $n\geq 4$....

Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $1$. The number $2$ appears onc...

Problem Find six positive integers $x_1, x_2, \dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do n...

Problem (1) Find a counterexample to Baillie-PSW primality test or prove that there is no one. Problem (2) Find a composite $n\equiv 3$ or $7\pmod{10...

Conjecture For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once. Equivalently: Conjecture For any prime $p>5$, $p^2$ doe...

Question Does there exist a $3\times 3$ magic square composed of distinct perfect squares?...

Conjecture Does a perfect cuboid exist?...