Why Fermat's little theorem

Question

History: read the statement of Fermat’s Little Theorem in a letter of $1640$. Why is it called “little” (relative to which “great” one)? (Hint: Fermat’s Last Theorem.)

Solution

Pierre de Fermat stated the result $a^{p-1}\equiv 1\pmod p$ in a letter of $1640$ (to Frénicle de Bessy), without proof: the first published proof is by Euler, almost a century later.

The adjective “little” serves to distinguish it from Fermat’s Last Theorem (also called the “great theorem”): the assertion that the equation $x^n + y^n = z^n$ has no positive integer solutions for $n>2$. The latter, jotted by Fermat in the margin of a book with the famous phrase “I have a marvellous proof that this margin is too narrow to contain”, remained a conjecture for over $350$ years, until Andrew Wiles’s proof in $1994$.

The “little” theorem is therefore an elementary and easily provable result; the “great” one was one of the most famous and difficult open problems in the history of mathematics.

Links

Topics: Probability distributions
Concepts: Modular arithmetic
Skills: Reasoning by cases
People: Pierre de Fermat
Exercise type: Word problem