Related papers: Note on the additive complements of primes

Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the…

Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.

We consider fuzzy valued functions from two parametric representations of $\alpha$-level sets. New concepts are introduced and compared with available notions. Following the two proposed approaches, we study fuzzy differential equations.…

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

In this paper, we derive a more precise version of the Strong Pair Correlation Conjecture on the zeros of the Riemann zeta function under Riemann Hypothesis and Twin Prime Conjecture.

We give necessary and sufficient conditions for the Chebyshev inequality to be an equality.

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

As a corollary to the recent extraordinary theorem of Maynard and Tao, we re-prove, in a stronger form, a result of Shiu concerning "strings" of consecutive, congruent primes.

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, we give an elementary proof of the additivity of the functional inverses of the resolvents of large $N$ random matrices, using recently developed matrix model techniques. This proof also gives a very natural generalization of…

In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins

In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are…

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.

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. S\'ark\"ozy and Szemer\'edi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved…

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…

We explicitly determine the associated primes of every power of a complementary edge ideal, prove that they satisfy the persistence property, and compute the $\text{v}$-function. In the course of the proofs, we completely describe the…

One of the two basic theorems in [5] on the existence of solutions of PDEs is improved with the use of a group analysis type argument.

We prove a refinement of the results of Gross and Zagier on prime factorizations of singular moduli.

In this supplementary appendix we provide additional results, omitted proofs and extensive simulations that complement the analysis of the main text (arXiv:1201.0224).

We offer some new applications of an extension of Abel's lemma, as well as its more general form established by Andrews and Freitas. A nice connection is established between this lemma and series involving the Riemann zeta function.