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So this question has strange origins: I was looking at the Leibniz series for $\pi$, and I started to wonder about the relationship between the partial sum and the parity of the value $n$ in the denominator: $2n+1$. (Since these partial sums are related to trig-functions, which ultimately help understand the proof of Euler's formula, I thought this question might be worth thinking about.)

In any case, is there anything particularly interesting about about the value of $n$ in odd-integers? That is, if for some odd number $2n+1$, are the interesting results if $n$ is even or $n$ is odd? (Mersennne primes sort of come to mind, but those are of the form $2^{p}-1$.)

Perhaps as a general direction, can we say something about whether a number is prime, is not prime, is near primes, given we know something about this $n$ (perhaps we could consider cases where $n$ is prime)?

Just wondering. Any thoughts, papers, or further reading would appreciated!

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closed as not a real question by William, Yuval Filmus, Noah Snyder, Norbert, J. M. Oct 10 '12 at 17:04

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

A.G., I can't follow your question. What do you mean by the value of $n$? Is the result "$2n$ is not prime unless $n=1$" something that would interest you? – Yuval Filmus Sep 17 '12 at 7:28
Looks like you're asking if there are any interesting differences between odd numbers of the form $4m+1$ and those of the form $4m+3$. Is that it? If so the answer is yes, plenty. – Marc van Leeuwen Sep 17 '12 at 11:55

If $n$ is even, then $2n+1=4m+1$ for some $m$; if $n$ is odd, then $2n+1=4m-1$ for some $m$.

This does have some interesting consequences, e.g., an odd prime is a sum of two squares if and only if it is $4m+1$. But it doesn't have the kind of consequence you have suggested, concerning primality; asymptotically, the number of $4m+1$ primes is the same as the number of $4m-1$ primes.

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