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 2 of 4: Primes one modulo four fail

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p\equiv1\pmod4\Longrightarrow b^2\equiv-1\pmod p\Longrightarrow b^3+b\equiv0\equiv0^3+0\pmod p

Detailed analysis

If a prime p greater than three is one modulo four, minus one is a quadratic residue. Choosing a nonzero b with square minus one makes the values at zero and b equal, so the map is not a permutation modulo p.