Browse problems

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 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…

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) =…

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 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 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 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 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…

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…