In fact, the only possible prime divisors of numbers that are equal to N*N+1, are the members of the sequence A002313.

Those are: 2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101 …

The number 2, as a divisor, covers 1/2 of them all. The number 5 covers 2/5 of those already covered by 2 and also 2/5 of those not divisible by 2. The number 13 covers 2/13 of them, and so on. Every prime P from this sequence, except for 2, covers (as a divisor) 2/P of all increased squares.

The infinite product (1/2)*(3/5)*(11/13)*(15/17)*(27/29)… tells us what percentage of N*N+1 numbers are primes.

If this infinite product converges to 0, which is to say that it diverges, which is the usual convention when we deal with infinite products … then the Landau’s problem isn’t solved. There may be a finite or an infinite number of primes which are squares increased by 1.

However, if this infinite product converges to something greater than 0, then Landau’s problem is solved and there are an infinite number of such primes. It is expected, that there actually are such primes all the way up. A positive percentage isn’t expected. That would be yooge!

Just prove the convergence of said infinite product! How hard could it be?

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http://lesswrong.com/r/discussion/lw/opg/open_thread_march_6_march_12_2017/do8b