History of Science and Mathematics Asked by Spencer on November 28, 2020

The Wikipedia Page on Fermat numbers states that $F_5$ was “fully factored” in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form

$$2^{n+1}k+1$$

for some $k$ (later improved by Lucas) and found the smaller factor of $F_5$, the prime $641$, by trial and error. What isn’t clear to me is whether Euler also verified that the larger factor $F_5 / 641 = 6,700,417$ was prime (it is).

C. Edward Sandifer’s book *How Euler did Even More* contains the passage

Euler did not speculate in print on whether the other factor, 6,700,417, is prime. It is prime, but

there is no evidence that Euler ever tried to find out.

but continues along the lines of how easy it is to prove, using the same tools Euler used to find the smaller factor.

You can’t say a number is “fully factored” until you know all of its factors are prime.

I was originally going to ask whether Euler proved $6700417$ was prime, but in light of the above passage, I guess we’ll never know.

Can we really say that $F_5$ was fully factored in 1732, or is there some date for the verification of $6700417$ that’s more appropriate?

*(user6530 pointed me to a cleaner version of the text than the Euler Archive preprint I’d originally linked to.)*

Once you know that factors of $F_5$ must be of the form $64k+1$, then it is very easy to try all of the possible factors of 6700417. You need try $64k+1$ for $kle 40$. You can easily use Sieve of Eratosthenes to eliminate at least some non prime numbers. For example eliminating numbers that are divided by 3,5,7,11 you left only with sixteen possible values for $k$: 3,4,7,9,10,12,15,18,22,24,25,30,33,37,40. It took for me less than two minutes to try one of the number using only paper and pencil and about half an hour to try all these numbers (hint: it is useful to use base 64 arithmetic). So, even Euler never claimed that 6700417 is prime, we can be almost sure that he knew it. And even if he didn't test it, it would be a simple exercise for his readers.

Answered by Alexei Kopylov on November 28, 2020

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