Find positive integers A, B, C, x, y, z with x, y, z \ge 3 and \gcd(A, B, C) = 1 such that A^x + B^y = C^z. Submit every value as a decimal string (no sign,…

Find a linear code over \mathrm{GF}(q) of length n and dimension k whose minimum distance d exceeds the best one known for those parameters. Bar to beat. The…

Find a sorting network on 18 inputs using fewer comparators than the best known. A sorting network is a fixed sequence of compare-exchange operations on pairs…

Find positive integers a, b, c such that all four of a^2 + b^2, \quad b^2 + c^2, \quad a^2 + c^2, \quad a^2 + b^2 + c^2 are perfect squares: a box with…

Arrange more non-overlapping unit spheres touching one central unit sphere in \mathbb{R}^{11} than the best known configuration. Equivalently: find unit…

Place 33 distinct points with integer coordinates in \{0, \dots, 10000\}^2 , no three on a line, such that no 7 of them are in convex position (the vertices…

With s(n) = \sigma(n) - n (the sum of the proper divisors of n ), find three distinct positive integers with s(n_1) = n_2, \quad s(n_2) = n_3, \quad s(n_3) =…

Status: solved (see Known). Kept as a template for matrix certificates. Find a 668 \times 668 matrix H with entries \pm 1 whose rows are pairwise orthogonal:…

Find a shorter schedule for Taillard's job-shop instance ta18 (20 jobs, 15 machines, 300 operations) than the best one known. Each job must pass through the…

Find a prime p with 2^{64} < p < 10^{24} and 2^{p-1} \equiv 1 \pmod{p^2}. Submit p as a decimal string (no sign, no spaces, no leading zeros): {"p":…

Find a short nonzero vector in a Darmstadt SVP Challenge lattice: shorter than the current Hall of Fame entry for that dimension, or the first entry in a…

Find an odd composite n with n \equiv \pm 2 \pmod 5 that passes both 2^{n-1} \equiv 1 \pmod n \qquad \text{and} \qquad F_{n+1} \equiv 0 \pmod n, where F_k is…

Find coprime positive integers a, b with c = a + b whose quality q = \frac{\log c}{\log \operatorname{rad}(abc)} > 1.63, where \operatorname{rad}(abc) is the…

Find a smaller Boolean circuit for the AES S-box: a straight-line program over AND, XOR and XNOR gates that maps the 8 input bits to the 8 output bits of the…

Losslessly compress enwik9 , the first 10^9 bytes of English Wikipedia, smaller than anyone has — counting the decompressor itself as part of the output. Bar…

Find integers x, y, z with x^3 + y^3 + z^3 = 114. Submit each integer as a decimal string (leading - allowed, no + , no spaces, no leading zeros): {"x":…

Find a permutation c_0, \dots, c_{32} of \{0, \dots, 32\} (a mark in row r , column c_r ) such that all displacement vectors between marks are distinct: for…

Find a nontrivial divisor d of the Fermat number F_{20} = 2^{2^{20}} + 1 , i.e. 1 < d < F_{20} with 2^{2^{20}} \equiv -1 \pmod d. Every divisor of F_{20} is…

Find 29 integers 0 = m_1 < m_2 < \dots < m_{29} \le 622 whose \binom{29}{2} = 406 pairwise differences m_j - m_i are all distinct (a Golomb ruler: no distance…

Find at most 72 words of length 6 over \{0, 1, 2\} such that each of the 3^6 = 729 words is within Hamming distance 1 of a chosen word (differs from it in at…

Colour each edge of the complete graph on the 43 vertices \{0, \dots, 42\} with one of two colours so that no 5 vertices have all \binom{5}{2} = 10 edges…

Find a 3 \times 3 magic square whose nine entries are distinct perfect squares. Give positive roots r_1, \dots, r_9 in row-major order, so the square is…

Place 2N = 142 points on the 71 \times 71 grid \{0, \dots, 70\}^2 so that no three of them lie on a common line, in any direction. Submit the points as [i, j]…

Find a quantum stabilizer code [[n, k, d]]_q — k logical qudits protected in n physical ones — whose distance d beats the best known for those parameters. Bar…

Colour the integers 1, \dots, 3704 with two colours so that no 7-term arithmetic progression a,\ a+s,\ a+2s,\ \dots,\ a+6s \qquad (s \ge 1) is monochromatic.…

Find a bilinear algorithm that multiplies two 4 \times 4 matrices over \mathrm{GF}(2) using fewer scalar multiplications than the best known, i.e. a shorter…

Find three 10 \times 10 Latin squares S_1, S_2, S_3 on the symbols \{0, \dots, 9\} (each symbol once in every row and every column) that are pairwise…

Find 2 \le k \le 6 positive integers t_1, \dots, t_k and a positive integer b with t_1^6 + t_2^6 + \cdots + t_k^6 = b^6. Submit decimal strings (no sign, no…

Find a shorter tour visiting all 1,904,711 cities of the World TSP instance than anyone has found. Bar to beat. Tour length < 7{,}515{,}755{,}912 , held by…

Find 157 lines on the points \{0, \dots, 156\} , each a set of 13 points, such that any two lines meet in exactly one point, every point lies on exactly 13…