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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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    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.2012-09-17

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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.