Problem 4

Determine all pairs

(n,p)

of positive integers such that

p

is a prime,

n

not exceeded

2p

, and

(p-1)^n+1

is divisible by

n^{p-1}

Step 4 of 5: Exclude primes larger than three

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In plain words

The first nonzero binomial term has exactly $p$ -adic order two.

(p-1)^p+1\equiv p^2\pmod {p^3}

Detailed analysis

For $n=p$ , the binomial expansion gives $(p-1)^p+1=(-1+p)^p+1=p^2+\text{a multiple of }p^3$ . For $p>3$ , divisibility by $p^{p-1}$ would imply divisibility by $p^3$ , contradiction.