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Register a new userCongruence Properties of Mersenne Primes
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In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer n} and generalize the result to all powers of 2. We also tabulate and show their relationship with other whole numbers up to 10. A number of minor results are also proved. Based on these results, approaches to determine the cardinality of Mersenne primes are discussed.
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We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $max_{nle x} sum_{p mid 2^n-1} 1/p$ to within $o(1)$ and $max_{nle x} sum_{dmid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $xtoinfty$. This refines, conditionally, earlier estimates of ErdH{o}s and ErdH{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $nle x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $sum_{pmid 2^n-1} 1/p$ and $sum_{dmid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
We obtain an upper bound for the number of pairs $ (a,b) in {Atimes B} $ such that $ a+b $ is a prime number, where $ A, B subseteq {1,...,N }$ with $|A||B| , gg frac{N^2}{(log {N})^2}$, $, N geq 1$ an integer. This improves on a bound given by Balog, Rivat and Sarkozy.
It is proven that there are infinitely prime pairs whose difference is no greater than 20.
Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from normal scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations.
By combining a sieve method of Harman with the work of Maynard and Tao we show that $$liminf_{nrightarrow infty}(p_{n+m}-p_n)ll exp(3.815m).$$
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