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Arithmetic Function Yielding Zeros Related to Sophie Germain Primes

Aram Tangboonduangjit


A prime number $p$ is said to be a Sophie Germain prime if $2p+1$ is also prime. The first few Sophie Germain primes are $2,3,5,11,23, 29, 41, 53$. Let $p$ be an odd prime. Define an arithmetic function $f_p: \mathbb{N}\to \mathbb{Z}_{2p+1}$ by $$f_p(n) = n^{n-1}+(-1)^{n+1}\cdot 2 \pmod{2p+1}.$$ We have found some curious properties of such function. In particular, we have that if $p$ is a Sophie Germain prime, then $$f_p\Big(\frac{p+1}{2}\Big) = 0.$$ The converse of this statement is also true, provided that $2p+1$ is not divisible by $3$ and $5$.

Last modified: 2016/12/21 16:07

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