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The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...

Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime. QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...