Mathematics Problem Archive

Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For...

For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$....

Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?...

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1....

Find all integer solutions to $n! + 1 = m^2$....

For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z....

For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$?...

Do Lychrel numbers exist in base 10?...

Is 10 a solitary number (no other number shares its abundancy index)?...

Does every nonnegative integer appear in Recamán's sequence?...

Do Lychrel numbers exist in base 10?...

Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?...

Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it ...

Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - ...

Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \le i \le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0...

Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?...

Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \dots + (n-1)^{n-1} + 1$, then $n$ is prime?...

In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of...

Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \le 2^k < n$?...

Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the dia...

Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?...

Does there exist a composite number $n \equiv 3$ or $7 \pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?...

Does there exist an even Fibonacci pseudoprime?...

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $ N \gg ...

Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?...

Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?...

Conjecture There is no odd perfect number....

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 $E/K$ 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....

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...

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

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

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 ...

Erdős: In the binary expansion of sqrt(2), are there arbitrarily long sequences of 0's? Can you find a single algebraic number with this property?...

Alon, Peres: Given any subset S of the integers modulo a prime p, what is the least K=K(p) for which there always exists an m so that mS has no gap of...

Niederreiter: Show that there exists a B so that, for every n > 0, there exists a k relatively prime to n whose continued fraction has partial quotien...

/Solymosi: Finite field Sylvester-Gallai: Suppose S is a tranversal of Z_(p)^(2), i.e., a set of points in the affine plane so that every row and colu...

Erdős-Turán: Suppose that S is a set of positive integers with the property that S+S -- that is, all sums of the form s_(1)+s_(2) for s_(1), s_(2) in ...

Olson: Every sequence of 2n-1 elements from a group of order n (written multiplicatively) has an n element subsequence with product 1 (in the given or...

Wills, Cusick: Suppose k runners having distinct constant speeds start at a common point and run laps on a unit length circular track. Then for any gi...

Erdős: Suppose that S is a set of positive integers with the property that no element is the sum of a nonempty set of other elements. Such a set is ca...

Dudeney: Is it possible to choose 2n points in an n by n grid in the plane so that no three are collinear? Conjecture: no. In fact, it is conjectured ...

Jaeger: If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y in F^(n) which haveal...

Graham: Is x^(2)+y^(2)=z^(2) partition regular? That is, is it true that every coloring of the positive integers by a finite number of colors contains...

Rado: Is it true that, for every n, there is an integer M(n), so that whenever a linear homogeneous equation in n variables is Ramsey (in the positive...

Erdős-Strauss : Is it possible, for each positive integer n, to find positive integers a, b, and c so that 4/n = 1/a + 1/b + 1/c ? See this....