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Old 2020-05-22, 13:19   #23
Dr Sardonicus
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Feb 2017

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Originally Posted by bhelmes View Post
if I regard only the primes p>=5 with p=x²+1 (x>1) and
the primes p with p | (x²+1 ) with p > x

can i derive any suggestion about the factorisation of p-1 in advance
(except divisible by 4 :-) )
Originally Posted by bhelmes View Post
if p=x²+1 then x | p-1

if p | (x²+1) ???

Can I calculate q=x²+1 with q=r*p and x and r known; then f | p-1 ; f ???
Originally Posted by bhelmes View Post
I have the factorisation of
f(n)=n²+1=r*p where r element of N and p is prime
It might help if you wouldn't keep trying to change the question.

In the first above-quoted post, you're asking whether knowing the x < p for which p divides x2 + 1 is of any help in factoring p-1. The answer is "No."

In the second above-quoted post, your hypothesis "x and r known" is nonsensical.

In the third above-quoted post, are assuming that n is given (this variable was named x in previous posts; so in addition to changing the question, you are also gratuitously changing your notation). And, apparently, you are now asking whether, knowing n and a prime factor p of n2 + 1, helps factor p - 1.

Again, the answer is "No." What you do know is that n (mod p) is one of the square roots of -1 (mod p); -n (mod p) is the other square root of -1 (mod p).

Knowing the square roots of -1 (mod p) can help find a and b such that p = a2 + b2. You could then check whether the condition I mentioned in this post is satisfied; and, if you're very lucky and it is satisfied, obtain a (hopefully non-trivial) factorization of p-1. But as I pointed out, the primes p for which the condition is satisfied are thin on the ground. And unfortunately, they appear to be thinner on the ground than primes p == 1 (mod 4) for which (p-1)/4 is actually prime. I'm guessing that p == 1 (mod 4), p <= X for which (p-1)/4 is prime have an asymptotic of order X/log2(X). The ones satisfying the condition I mentioned before, I have no idea, except numerical data indicate a smaller asymptotic.

Perhaps someone who knows the subject better than I could indicate what is known for the proportion of primes p == 1 (mod 4) for which the largest prime factor of p-1 is greater than pc where 0 < c < 1.
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