Of course you mean numbers of the form 2^(2^n) + 1; without the increment, primality is easy to check. :-) Anyway, are you sure that it would be so easy to check primality of 15-decimal-digit numbers without even a pocket calculator, especially with no guarantee that there was any reason to do so? (Notice, for example, that you needed to add "if someone told me to".)

For more on this, see KConrad's comment https://matheducators.stackexchange.com/questions/16636/is-t... and kcrisman's comment https://matheducators.stackexchange.com/questions/16636/is-t... on the post linked by your parent.

Yes, thanks for the correction :) I've fixed it above.

2^(2^5) + 1 is 4296967297 = 641 * 6700417. Not at all a huge number today, but in those days it'd probably be considered "astronomical".

Also, the series 2^(2^n)+1 grows extremely quickly; the next number is 18446744073709551617 = 274177 * 6728042131072, and after that 340282366920938463463374607431768211457 = 59649589127497217 * 5704689200685129054721.

2^32+1 is likely reaching the limits of what an elementary school child is able to do by hand. As for the larger numbers: they do get unwieldy quickly, but I think it might be possible to manually find the factors of them with probabilistic primality testing.

Well, the child will be bored much earlier :-)

For Fermat numbers 2^(2^n) + 1, it is well known that every prime factor should have the form k 2^(n+2) + 1. The sequence (A007117) thus goes like: k = 5 for n = 5 (Euler in 1732, very managable), k = 1071 for n = 6 (Thomas Clausen in 1854) and k = 116503103764643 for n = 7 (only known in 1970 after the modern computer).

Just a typo but 2^(2^5) + 1 is 42946967297, you skipped the fourth digit.

Wait, you expect to be able to easily check primality of 2^32 + 1 by hand? Like with a naive algorithm you’d need to use erastothenes sieve to find primes up to 2^16 to test against. Granted, 641 itself is rather small but if you didn’t expect to encounter a factor that early...

> Like with a naive algorithm you’d need to use erastothenes sieve to find primes up to 2^16 to test against.

π(2^16)=6542: if you told me in elementary school that I might disproving something famous I can see myself working on that for a couple of hours daily for a week or so.

What if I told you you'd very probably just show that the pattern continues to hold for the first 5 n?

You’d stifle mathematical innovation: after all, there are quite a few things that have been proved or discovered by people who were fortunate enough to not be told that they were solving a hard problem.

There were so many other more interesting mathematical results to work on back then, that were equally accessible to a bright and motivated schoolchild. Euler and Fermat had it easy!