Unsolved Problems

Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for...

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

Problem Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $$ m(A) = - \min_x \sum_{a \in A} \cos(ax). $$ What is $m(n) = \min_A...

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational. Conjecture There doe...

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) = ...

Conjecture $\pi$ and $e$ are algebraically independent...

Conjecture There is no odd perfect number....

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

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...

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 Let $E/K$ be an elliptic curve over a number field $K$. Then the order of the zeros of its $L$-function, $L(E, s)$, at $s = 1$ is the Morde...

Let $B \subseteq {\mathbb N}$. The representation function $r_B: {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j...

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

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

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the co...

Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, then the extension field...

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 For any $\epsilon>0$ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$ Since $\epsilon$ can be replaced by a smaller ...

Question Is Euler-Mascheroni constant an transcendental number?...

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

Conjecture Are all Fermat Numbers $$ F_n = 2^{2^{n } } + 1 $$ Square-Free?...

Conjecture A Fermat prime is a Fermat number $$ F_n = 2^{2^n } + 1 $$ that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_...

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?...

Conjecture A Mersenne prime is a Mersenne number $$ M_n = 2^p - 1 $$ that is prime. Are there infinite number of Mersenne Primes?...

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 ...

Say that a set $S \subseteq {\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums. Conjecture There exists a fixed cons...

Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$. Assume we start with some number $n$ and repeatedly take ...

Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1....

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...

For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elemen...

Conjecture Let $G$ be an abelian group of odd order and let $A,B \subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered...

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?...