4 ms·
I thought it was wrong -- but looking back at Euclid's proof, Euclid doesn't specify that p_1...p_n be all the primes -- just all the known primes, so there cou
by glial 7y ago
I thought it was wrong -- but looking back at Euclid's proof, Euclid doesn't specify that p_1...p_n be all the primes -- just all the known primes, so there could be gaps. He says that either
(1) -- p_1×p_2×p_3×…×p_n +1 is prime, or
(2) -- p_1×p_2×p_3×…×p_n +1 is not prime, but it's also not divisible by any of the pre-specified primes, so there must be another prime that divides into it aside from p_1...p_n.
See here: https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/propI...
- _Microft 7y agoOK, I understand. Strange that the non-successive primes are numbered with successive indices though. PS: Well, actually not because they are not numbering all primes but numbering primes they know.
- kosievdmerwe 7y agoEven if they are successive then the product plus one might not be prime, since the product plus one is so much larger than p_n, there might be a prime that divides it (and likely at least 2). The first non-prime example is 2x3x5x7x11x13 + 1 which equals 59 x 509
- _Microft 7y agoSure, see my first post about that. I amended it shortly after I posted but it felt wrong to remove the part that people responded too already.