21 votes
Accepted

Sum of reciprocals of Sophie Germain primes

Here is a general result. For a sequence of nonnegative numbers $\{a_n\}$, let $A(x) = \sum_{n \leq x} a_n$. For example, if $S \subset \mathbf Z^+$ and we set $a_n = 1$ when $n\in S$ and $a_n = 0$ ...
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21 votes

Sum of reciprocals of Sophie Germain primes

Googling "sum of reciprocals of Sophie Germain primes" brings up the very recent paper: Wagstaff, Samuel S. jun., Sum of reciprocals of germain primes, J. Integer Seq. 24, No. 9, Article 21....
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15 votes
Accepted

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

On the Riemann hypothesis and the difference between primes by Adrian W. Dudek states the result (Theorem 3, at least in the arXiv version) that any $C>1$ works (for $n$ sufficiently large).
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7 votes

Reference request: Numbers composed of given primes

As Mikhail Tikhomirov commented, the denominator should have $\log$ of $p_j$ instead of $p_j$ itself. For a reference see Theorem 5.3, and more generally, section 2 (`The geometric method') of Chapter ...
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5 votes

Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$

It's often the case with such tests that the "only if" part is more or less easy to prove, while the "if" part is inaccessible for proving or disproving. Below I prove the "...
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3 votes
Accepted

A specific Diophantine equation restricted to prime values of variables.

Pace Nielsen and Cody Hansen just put this preprint on the Arxiv which shows that no triple threats exist.
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2 votes

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

I wanted to give some additional remarks to Will Sawin's answer and the associated comments. Carneiro, Milinovich, and Soundararajan certainly have the best result in literature (as far as I'm aware). ...
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