hello number theorists ,
I'm beginner and I have troubles with this lemma which i belive it is true ,
let a prime number greater than , and let be the set of primes less than , show that there exist from , such that divides
Thanks in ADvance
Maybe this will help you!
I find similarity between the problems... Euclid's Proof of the Infinitude of Primes (c. 300 BC)
TO Jamix , what about ? , anyway here is the solution
( I apologize , I forgot to mention that A contains at least 3 elements )
First of all we shall prove that A is infinite , indeed suppose it is finite for sake of contradiction, consider , let , by conditions has all prime divisors in A, but is prime with each element of Except , we conclude that for some positive integer , that is ,
now it is easy to prove that are in A , in particuler and , for some positive integers since we conclude that , looking we get that is which is is a contradiction ,
let q a prime number , since A is infinite , there exist at least q-1 prime number which have the same residue in , we can suppose that this residue isn't 0 , let those primes , hence by Fermat little theorem , we have and we are done