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A Fermat prime is a Fermat number that is also a prime number:The defining form alone does not imply primality. For example, $65537$ is a Fermat prime, while $4294967297$ is a Fermat number with proper divisors. For $k\ge 1$, every Fermat prime is one modulo four and two modulo three. Pépin's test gives an exact primality testing criterion for these candidates, rather than relying on their form.

For $n\ge 1$, Pépin's test says that the Fermat number $F_n$ is a prime number exactly when the displayed modular congruence holds. Modular exponentiation evaluates it by exponentiation by squaring without constructing the enormous ordinary power. The index restriction excludes $F_0=3$, where the base itself is divisible by the candidate.^[ref]

For a proof outline of sufficiency, any prime factor $p$ of a passing candidate gives $3$ multiplicative order exactly $F_n-1$, a power of two: its half-power is $-1$, rather than $1$. Fermat's little theorem forces $F_n-1\mid p-1$, so $p\ge F_n$, and hence $p=F_n$. Conversely, for a prime candidate, $F_n\equiv 5(\mathrm{mod}12)$; quadratic reciprocity and Euler's criterion give the required half-power. At $n=1$, the check is $3^2\equiv 4=-1(\mathrm{mod}5)$.

By Carl Pomerance. Slides for the ICERM meeting "Seventy-five years of Mathematics of Computation", 1–3 November 2018.

Online text: math.dartmouth.edu/~carlp/mathcomptalk2.pdf

Cited passages: slide 14 states Pépin's test and outlines its proof; slides 15–16 describe Fermat numbers and their role in integer factorization; slide 18 gives the Gauss-Wantzel theorem and distinguishes Gauss's construction result from Wantzel's necessity proof. Numerical records elsewhere in these 2018 slides are historical records, not claims about current records.