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Results tagged with analytic-number-theory
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user 2480
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
8
votes
Accepted
Polynomial whose values divide $n!$
Here is a completely elementary proof, inspired by Pasten's comments.
Let $P(n)=an^2+bn+c$.
Take $n=a^5x^4+2a^3(ab+2a+1)x^3+a(2a^3c+a^2b^2+6a^2b+3ab+6a^2+5a+1)x^2+(ab+2a+1)(2a^2c+2ab+b+2a+1)x+a^3c^2+2 …
- 5,103
9
votes
Prime square offsets: Why is +7 more frequent than -7?
Modulo 6 the squares are 0,1,4,3,4,1 and the squares+7 (or -5) can only be 1,2,5,4,5,2, of which 3/6 can at all be prime. The squares-7 (or +5) are 5,0,3,2,3,0 of which only 1/6 can be prime. Obviousl …
- 5,103
11
votes
2
answers
726
views
What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?
Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges?
Can one compute a few digits based on euristic considerations or plausible c …
- 5,103