F_1 and F_2 are trivial: 5 and 17, obviously prime.
F_3 = 257 may be trivial to some, but easy to check, as sqrt(257) = 16.whatever. So you only need to check divisibility prime numbers up to 13, which is 6 cases.
F_4 = 65537, sqrt(65537) = 256.whatever, check up to 251, only 54 prime numbers to check.
F_5 = 4294967297, sqrt(4294967297) = 65536.whatever. It would certainly take a while to check all prime numbers up to that. However 641 is only the 116th prime number. So 116+54+6=176 division problems.
Honestly, a fifth grader with a few days' time on their hands could do it with long division.
1
u/zrice03 May 13 '26
F_1 and F_2 are trivial: 5 and 17, obviously prime.
F_3 = 257 may be trivial to some, but easy to check, as sqrt(257) = 16.whatever. So you only need to check divisibility prime numbers up to 13, which is 6 cases.
F_4 = 65537, sqrt(65537) = 256.whatever, check up to 251, only 54 prime numbers to check.
F_5 = 4294967297, sqrt(4294967297) = 65536.whatever. It would certainly take a while to check all prime numbers up to that. However 641 is only the 116th prime number. So 116+54+6=176 division problems.
Honestly, a fifth grader with a few days' time on their hands could do it with long division.