Exercise 2193Number theory › PrimesM1Added 3 Nov 2026

The Fermat quotient

#Fermat quotient#prime#congruence

Statement

Define the Fermat quotient $q_p(a)=(a^{p-1}-1)/p$ for $p\nmid a$ and compute $q_5(2)$ .

Hint

Compute $2^4-1$ and divide by $5$ .

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Definition. $q_p(a)=(a^{p-1}-1)/p$ ; this is an integer by Fermat's little theorem.

Compute. $q_5(2)=(2^4-1)/5=15/5=3$ .

Properties. $q_p(ab)\equiv q_p(a)+q_p(b)\pmod p$ ; $q_p(a^k)\equiv k\,q_p(a)\pmod p$ .

Conclusion. $q_5(2)=3$ .

Application. Wieferich primes satisfy $q_p(2)\equiv0\pmod p$ .