arXiv subjects
Publications and source records attributed to Paolo Starni.
P\'epin's primality test asserts that, for $n\ge2$, \[ 3^{(F_n-1)/2}\equiv -1 \pmod{F_n} \] if and only if $F_n$ is prime. We establish a natural analogue of P\'epin's criterion for pseudoprimality. More precisely, we prove that, for \mbox{$n\ge5$,} \[ 3^{(F_n-1)/2}\equiv 1 \pmod{F_n} \] if and only if $F_n$ is pseudoprime to the base $3$.
We answer the question positively. In fact, we believe to have proved that every even integer $2N\geq3\times10^{6}$ is the sum of two odd distinct primes. Numerical calculations extend this result for $2N$ in the range $8-3\times10^{6}$. So, a fortiori, it is shown that every even integer $2N>2$ is the sum of two primes (Goldbach conjecture). Of course, we would be grateful for comments and objections.
The multiplicative structure of an odd perfect number $n$, if any, is $n=π^αM^2$, where $π$ is prime, $\gcd(π,M)=1$ and $π\equiv α\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv π\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $π$). We further extend the proof to $x=α+2$, $α+2$ prime, with the restriction that the congruence modulo $4 x$ does not include $n$. Besides, we note that the first extension of Touchard's result holds also with an exclusive disjunction, so that $π\equiv 1\pmod{12}$ is a sufficient condition because $3\nmid n$.
The Euler's form of odd perfect numbers, if any, is $n=π^αN^2$, where $π$ is prime, $(π,N)=1$ and $π\equiv α\equiv 1 \pmod{4}$. Dris conjecture states that $N>π^α$. We find that $N^2>\frac{1}{2}π^γ$, with $γ=max\{ω(n)-1,α\}$; $ω(n)\geq 9$ is the number of distinct prime factors of $n$.