Problem 3

Find all positive integers

n

such that for any integer

k

there exists an integer

a

for which

a^3+a-k

is divisible by

n

Step 1 of 4: Powers of three work

Previous step Next step

(a^3+a)-(b^3+b)=(a-b)(a^2+ab+b^2+1)

Detailed analysis

For modulus three to a positive power, suppose the two values coincide. The displayed factorization holds, and the second factor is never divisible by three, as a direct check modulo three shows. Hence a and b are congruent, so the map is injective on a finite residue system and therefore a permutation. Modulus one also works.