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For a prime number $p$ and an integer $a$ not divisible by $p$, Fermat's little theorem states the modular congruenceEquivalently, for every integer $a$, including $a<0$ and multiples of $p$,The first formulation follows from the second by multiplying by a modular multiplicative inverse of $a$. Conversely, multiply the first by $a$ when $p\nmid a$; when $p\mid a$, both sides of the second are zero modulo $p$. The divisibility condition in the first formulation is essential.^[ref]

For $p=7$, $3^6=729=7\cdot 104+1$, so $3^6\equiv 1(\mathrm{mod}7)$ and $3^7\equiv 3(\mathrm{mod}7)$. Also $(-2{)}^6=64\equiv 1(\mathrm{mod}7)$, while $7^6\equiv 0(\mathrm{mod}7)$: a zero residue class cannot satisfy the first formulation. It does satisfy $7^7\equiv 7\equiv 0(\mathrm{mod}7)$. A failure at a zero residue class therefore says nothing against a candidate's being a prime number.

The prime number $p=2$ is included. Its only nonzero residue class is one: every odd integer $a$ satisfies $a^{2-1}=a\equiv 1(\mathrm{mod}2)$. Every integer also satisfies $a^2\equiv a(\mathrm{mod}2)$, because $a(a-1)$ is even. Testing the first formulation with an even integer would violate its hypothesis.

The useful surprise is that every nonzero residue class returns to one at the same exponent $p-1$, although its shortest return can be earlier. This gives exponent reduction for units: for a nonnegative integer $k$, $a^k\equiv a^{k\mathrm{mod}(p-1)}(\mathrm{mod}p)$ when $p\nmid a$. It also computes a modular multiplicative inverse as $a^{p-2}$ modulo $p$, including $p=2$. Repeated squaring makes these modular exponentiations practical.

In group theory, the nonzero residue classes form the multiplicative group of integers modulo n with $n=p$, of group order $p-1$. Lagrange's theorem says that the multiplicative order of $a$ divides $p-1$, giving the guaranteed return. Euler's theorem extends this to a general integer modulus $n$, using the units and Euler's totient function $\phi (n)$; the Carmichael function gives the least common return exponent for all units. A nonzero residue class modulo a composite number need not be a unit.

Leonhard Euler's paper presented in 1732 and published in 1738 states a historical formulation: Theorem 1 says that powers with exponent $n-1$ leave zero or one upon division by a prime number $n$. The preceding discussion states the difference-of-powers version and explicitly says that he does not have its proof.^[ref]

A Fermat number is the positive integer $F_n=2^{2^n}+1$, indexed by a nonnegative integer $n$. The first five are $3,5,17,257,65537$, all prime numbers; the next is the composite number $F_5=4294967297=641\cdot 6700417$. This family offers a compact lesson in number theory: a very persuasive finite pattern can fail, while an exact divisibility relation settles a gigantic case with little computation.

Why does the exponent itself have the form $2^n$? Suppose $m=ab$ is a positive integer with an odd integer factor $b>1$. The polynomial factorization $x^b+1=(x+1)(x^{b-1}-x^{b-2}+\cdots -x+1)$ shows that $2^a+1$ is a proper divisor of $2^m+1$. Consequently, if $2^m+1$ is a prime number, $m$ must be a power of two. That is a necessary condition, rather than a promise that the surviving candidates are prime.

The Fermat number product identity makes distinct Fermat numbers coprime integers. Thus every term has a prime factor absent from all earlier terms, whether or not that term is itself a prime number. This gives an explicit route to infinitely many different prime numbers, without establishing that infinitely many terms of this particular sequence are prime. Meanwhile the prime divisor of a Fermat number criterion filters possible factors through a small multiplicative order calculation.

The family also links arithmetic to Euclidean geometry: a constructible regular polygon can have a number of sides equal to a power of two times a product of distinct Fermat primes. The Gauss-Wantzel theorem gives the precise criterion. For instance, $15=3\cdot 5$ and $17$ qualify, but $9=3^2$ does not. The word "distinct" matters. Carl Pomerance describes this connection in Primality testing: then and now by Carl Pomerance (slide 18).

Historically, Leonhard Euler records Pierre de Fermat's proposed primality pattern, then refutes it by finding the factor $641$ of $F_5$. His paper was presented in 1732 and published in 1738; those are different events, not competing dates for one publication.^[ref]

A short modern check needs only two representations of $641$:In modular arithmetic these give $5^4{2}^{28}\equiv 1$ and $5^4\equiv -2^4(\mathrm{mod}641)$. Substitution yields $2^{32}\equiv -1(\mathrm{mod}641)$, so $641\mid F_5$. This verifies the factor directly; it does not claim to reconstruct Euler's search process.

Translated Observations on a certain theorem of Fermat and on others concerning prime numbers by Leonhard Euler into English.