Question

prove that there are infinitely many primes of the form 8k-1​

Answer 1

Answer:

There exist infinitely many primes of the form 8k-1, where k = 1, 2, ... and, in view of property IV, p must be of the form 8k+1. ... We have thus proved that for any natural number n >1 there exists a prime p greater than n that is of the form 8k-1.

Answer 2

Step-by-step explanation:

Let p1,p2,…,pk be the list of ALL primes of the form 8s+7. Let

N=(p1p2⋯pk)2−2.

Note that N≡7(mod8) and is odd. If p is a prime that divides N, then

(p1p2⋯pk)2≡2(modp).

Thus

(2p)=1.

Thus p≡±1(mod8).

So all primes that divide N must be of the from 8s+1 or 8s+7. But not all of them can be of the form 8s+1 (ask why???)

So there must be one of the form q=8s+7. Now see if you can proceed from here.

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