MathDB

Everything is sum of two cubes...

Source: Ukrainian TST 2007 problem 6

June 7, 2006

geometry3D geometrynumber theory proposednumber theory

Problem Statement

Find all primes

p

for that there is an integer

n

such that there are no integers

x,y

with x^3 \plus{} y^3 \equiv n \mod p (so not all residues are the sum of two cubes). E.g. for p \equal{} 7, one could set n \equal{} \pm 3 since

x^3,y^3 \equiv 0 , \pm 1 \mod 7

, thus x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7 only.