Problems & Puzzles: Conjectures

Conjecture 5. Are there infinitely many primes of the form n2+1?

It’s believed that there are infinite primes of the form n2+1.

Even more than that, Hardy & Littlewood guessed that the number of such primes less than n, P(n), was asymptotic to c.sqrt(n)/ln(n).

And it happens that c~ 1.3727 =P {1-[(-1)(p-1)/2]/[p-1]}, where the product is taken over all the odd primes. (Ref. 2, pp. 4-5)

Solution

Daniel Gronau wrote (8/10/01):

Here are some basic results to conjecture No. 5 ("The number of primes p = n²+1 is not limited").

We are interested in odd numbers only, so let q(n) = 4n²+1. So this is true:

1) p | q(n) <-> p = 4m+1, n = +-sqrt(m) (mod p)

2) gcd(q(n), q(n+1)) = 1

3) q(n) | q(m) <-> m = +- n (mod q(n))

A consequence from 1) is, that the chance of p | q(n) (given a prime p = 4m+1) is 2/p. This can lead to new approximations for the number of primes in the sequence q(n).

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