Related papers: A Conjecture on Primes and a Step towards Justific…

In this paper, we prove a few lemmas concerning Fibonacci numbers modulo primes and provide a few statements that are equivalent to Wall-Sun-Sun Prime Conjecture. Further, we investigate the conjecture through heuristic arguments and…

In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…

In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…

An technically interesting proof of a known theorem.

I give some claims on primorial prime numbers for interested readers in number theory.

We introduce a new semantics for justification logic based on subset relations. Instead of using the established and more symbolic interpretation of justifications, we model justifications as sets of possible worlds. We introduce a new…

The Twin Prime conjecture states that there are infinitely many pairs of distinct primes which differ by $2$. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang…

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers.…

Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related…

In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

In this note we prove an inequality involving primes and the product of consecutive primes.

Fairness assumptions are a valuable tool when reasoning about systems. In this paper, we classify several fairness properties found in the literature and argue that most of them are too restrictive for many applications. As an alternative…

In this paper, we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop a series of steps to prove the binary Goldbach conjecture in full.…