Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}} Conjecture: M_{q}\text{ is a prime iff: } \ S_{q-1} \equiv S_{0} \pmod{M_{q}} \text{ and iff: } \prod_{0}^{q-2} S_i \equiv 1 \pmod{M_{q}} … Read more

Behavior of the “mean prime factor” of numbers

This question concerns the behavior of a function f() that maps each number in N to its mean prime factor. I previously posted premature questions, now deleted, which explains the cites below to several who contributed observations. Also, the question, “Distribution of the number of prime factors,” may be relevant. Define f(n) to be the … Read more

A problem on prime numbers

Given integers a,b,c,d∈[2n,2m] with m>n>1, how many primes p are there in [nα,nβ] for some 1<α<β such that 0<amod 0<b\bmod p<n^{\alpha/k} 0<c\bmod p<n^{\alpha/l} 0<d\bmod p<n^{\alpha/l} holds where k,l>2 is fixed? Assume n,m,\alpha,\beta,k,l are fixed. Heuristically we can solve for average case of choice of a,b,c,d following way: \frac{|\{p:a\bmod p<n^{\alpha/k}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{|\{p:b\bmod p<n^{\alpha/k}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{n^{\alpha/k}}{(n^{\beta}-n^\alpha)/\log(n^{\beta}-n^\alpha)} \frac{|\{p:c\bmod p<n^{\alpha/l}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{|\{p:d\bmod p<n^{\alpha/l}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{n^{\alpha/l}}{(n^{\beta}-n^\alpha)/\log(n^{\beta}-n^\alpha)} So \mathsf{Prob}(a\bmod … Read more

Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let’s introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let’s now write $\mathcal{G}(x):=\sum_{k\leq x}\left(\dfrac{G_{2}(k)}{G_{1}(k)}-1\right)$. Does this last series diverge? Converge towards a positive constant? Many thanks in advance. Answer AttributionSource : Link , Question Author : Sylvain JULIEN , Answer Author : Community

Large Gaps Between Almost Primes

What is the best lower bound for the longest interval contained in [1,x] free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the Westzynthius-Erdos-Rankin problem of finding large gaps between primes, as in this recent paper. In this question, it seems … Read more

Asymptotic estimate for a random model of primes

Question Let πrmc(x)=∑n≤x(n+a,P(√n))=11−1, where P(x) is the product of all primes less or equal to x and a is a random integer constrained to those values such that (n+a,P(√n))≤n for all n≤x. What is the asymptotic behaviour of the expected value of πrmc(x)? Below follows some background for the question, a possible partial approach, and … Read more

Effective prime number theorem

The prime number theorem implies that for every ϵ>0, there is n_\epsilon such that for all n≥n_\epsilon the number of primes in [n,cn] is at least \frac{(c−1−\epsilon)n}{\log n} and at most \frac{(c−1+\epsilon)n}{\log n} for every c>1. What is the precise value of n_\epsilon as a function of \epsilon known unconditionally and conditionally on reasonable conjectures? … Read more

The largest primes in the monster group construction

The last three prime numbers in the factorization of the order of the Fischer Griess friendly monster group are 47, 59, 71. ( On the other hand, the monster group can be contrcuted from the automorphism group of the Griess algebra which has 196884-dimensions. The Monster fixes (vectorwise) a 1-space in this algebra and acts … Read more

Are there any results about this higher degree Titchmarsh divisor problem?

Does there exist an asymptotic formula for $\sum_{p\le x}\tau(p−1)^n$ ? Here $n$ is an arbitrary positive integer and $\tau$ is the divisor function. The case of $n=1$ was done by Linnik, but when $n$ is big the question seems to be open. Answer AttributionSource : Link , Question Author : user97495 , Answer Author : … Read more

Prime powers between xx and x+xθx+x^\theta

By the result of Baker, Harman, Pintz (, for any sufficiently large x the interval [x−x21/40,x] contains a prime number. This result implies the asymptotic p(x)=x+O(x21/40) where the function p(x) assigns to each real number x the smallest prime number p≥x. Question. For which smallest possible constant θ is it known that [x−xθ,x] contains a … Read more