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Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...

Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a<p$ and $ m \ | \ \phi(n)$ then $n$ is prime. This question is a ...

For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$. Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$. Set $\hat{\phi}(1) = \{x_1\}$ and ...

Posting in MO since it was unanswered in MSE. Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...

Let $f\in\operatorname{int}(\mathbb Z)$, the ring of integer-valued rational polynomials. Define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments....