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Miss and Mister cassoulet char's user avatar
Miss and Mister cassoulet char's user avatar
Miss and Mister cassoulet char's user avatar
Miss and Mister cassoulet char
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theorem there is the following sieve :

Let $p,k>1,c,d$ be integers then we have : $$p=4k^2+1 \operatorname{is prime iff} p\neq c^2+d^2,c>1,d>1$$

$\implies $ https://math.stackexchange.com/questions/719700/if-a-prime-can-be-expressed-as-sum-of-square-of-two-integers-then-prove-that-th

The other case :

Partial answer see : On numbers which are the sum of two squares - The Euler Archive http://eulerarchive.maa.org/docs/translations/E228en.pdf

For example :

If someone ask you to find quickly if $257$ is a prime number use the sieve above .

https://en.m.wikipedia.org/wiki/Pierpont_prime $v=0$

As the sum is symmetric and homogeneous we need less term than in taking divisor and square roots of the prime.

The inverse of the prime counting function is roughly :

$$\pi^{-1}(x)\simeq p\left(x\right)=\sqrt{\ln\left(\ln\left(x!\right)\right)}\frac{a}{\sqrt{2}}x\int_{0}^{\frac{a}{\sqrt{2}}x}\frac{\left(\ln\left(a^{2}+1\right)+\ln\left(y^{2}+a^{2}\right)\right)}{\ln\left(\left(a^{2}+y^{2}-1\right)!\right)\cdot f\left(0\right)}dy-\ln\left(\ln\left(x!\right)\right),f\left(x\right)=\frac{\left(\ln\left(a^{2}+1\right)+\ln\left(\frac{a^{2}}{2}x^{2}+a^{2}\right)\right)}{\ln\left(\left(a^{2}+\frac{a^{2}}{2}x^{2}-1\right)!\right)},a=2$$

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