Five Primes and Then Never Again

Start with 2. Square it to get 4, square that to get 16, then 256, then 65,536, and so on: each step squares the previous power of two. Add one to each of those and you have the Fermat numbers $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$, $F_4=65,537$ and $F_5=4,294,967,297$.

The streak view gives each of them a verdict by dividing. To decide that 257 is prime, the laboratory divides it by every whole number from 2 up to 16, because a divisor larger than the square root would have to be paired with a smaller one. Nothing is left over in any of those divisions, so 257 has no divisor except 1 and itself.

Three, five, seventeen, 257 and 65,537 are all prime, and they are the first five cases of a rule that generates numbers with no obvious reason to be prime at all. Fermat looked at that run and conjectured that every number of this form is prime.

It is worth sitting with how reasonable that is. Five consecutive confirmations, no near miss, and the next case already too large to divide by hand. The mistake is not in the arithmetic, which is correct, but in what a finite run of cases is allowed to establish. This page separates the two: the five verdicts on the ladder are proofs about five numbers, and the conjecture they suggested is a statement about infinitely many.

$F_5$ is 4,294,967,297. Testing every odd number up to its square root means about thirty thousand divisions, which is why the sixth case was out of reach for a long time.

The way in is a theorem about what a factor can look like. Any prime that divides $F_n$, for $n\ge 2$, is one more than a multiple of $2^{n+2}$. For $n=5$ that multiple is 128, so the only candidates are 129, 257, 385, 513, 641, and so on. The fifth of them divides. That is the whole factor hunt: five divisions instead of thirty thousand, and the reason the view can show every one of them.

Euler had the weaker version of this theorem, with $2^{n+1}$ in place of $2^{n+2}$, which gives twice as many candidates and still finds 641. The sharper form is Lucas's, and it is the one the laboratory uses.

The Fermat numbers are

They satisfy $F_0{F}_1\cdots F_{n-1}=F_n-2$, from which any two of them are coprime: a common divisor of $F_m$ and $F_n$ with $m<n$ would divide both $F_n$ and $F_n-2$, hence divide 2, and every Fermat number is odd.

$F_0$ through $F_4$ are prime. $F_5=641\times 6,700,417$. Every prime factor of $F_n$ for $n\ge 2$ has the form $k\cdot 2^{n+2}+1$. Every $F_n$ for $5\le n\le 32$ is known to be composite, which is a result of other people's computation rather than of anything on this page. Whether any Fermat number beyond $F_4$ is prime is unknown.

The first five verdicts are proved by exhaustion, and the exhaustion is genuine. To settle whether $F_4=65,537$ is prime, it is enough to test the divisors up to $\sqrt{65,537}<256$. The laboratory tests all of them. Every possible case is covered, so the check is a proof of that one statement. This is the situation in which a finite computation really does prove something, and it is worth naming, because the same laboratory refuses the same verdict two rungs later.

The product identity has a two-line proof. $F_n-2=2^{2^n}-1=(2^{2^{n-1}}-1)(2^{2^{n-1}}+1)=(F_{n-1}-2)F_{n-1}$, and induction from $F_1-2=F_0$ gives the full product. Coprimality follows as above, and it yields a proof that there are infinitely many primes: each Fermat number contributes a prime factor that no earlier one has.

The Lucas form is cited rather than proved here. The shape of the argument is that if a prime $p$ divides $F_n$ then 2 has order $2^{n+1}$ modulo $p$, so $2^{n+1}$ divides $p-1$; the extra factor of two in the sharper form comes from 2 being a quadratic residue modulo $p$ when $n\ge 2$.

Gauss found the reason these particular primes matter beyond their own list. A regular polygon with an odd number of sides can be constructed with ruler and compass exactly when that number is a product of distinct Fermat primes. Three, five, 17, 257 and 65,537 sides are all constructible, and so is a 15-gon, since $15=3\times 5$. A regular 7-gon is not. Gauss found the 17-gon construction at nineteen, and the discovery is what turned him to mathematics. So the list of Fermat primes is also the list of odd prime numbers of sides a Greek geometer could have drawn, and it is a list nobody knows the end of.

What is known now: no Fermat prime beyond $F_4$ has been found, and every index from 5 to 32 has been shown composite. The frontier in the growth view is that result, and it belongs to the people who computed it, not to this page. Beyond it the status table records indexes where a factor is known, indexes shown composite with no factor in hand, and indexes about which nothing is known at all.

The counting heuristic suggests the five known Fermat primes are all of them, and a heuristic is not a proof in either direction. It is not known whether infinitely many Fermat numbers are composite either. A rule this simple, with an answer known for five of its cases and unknown for the rest of them, is the ordinary condition of the subject rather than an unusual one.