Some Problems in Elementary Number Theory and Modular Forms

2019

This paper deals more generally with @-numbers defined as follows. Call `\textit{alpha number} of order $k$', (denote its family by @$_{k;\mathbb{N}}$) any positive integer $n$ satisfying $\mathit{f}_k(n):=(\alpha_1/\alpha_2)n $ with $\mathit{f}_k(n):=\lfloor|\sigma_k(n)|\rfloor$, arbitrary pair integers $\alpha_1,\alpha_2$ is such that $1<\alpha_1,\alpha_2\leq \tau (n)$ where $\tau (n)$ is the number of factors of $n$, and $\sigma_k(n)$ is the sum of divisors function of $n$. We give some examples and conjecture that there is no odd alpha number of integral order above 1, which implies that there is no odd perfect, multiperfect or Ore's harmonic number greater than 1. In this paper, using Rossen, Schonfield and Sandor's inequalities, in addition to the aforementioned definition, we also provide a form for odd @-numbers, and remark that this form can be improved towards solving the conjectures of this paper. Some areas for future research are also pointed out as recom...

Journal of Number Theory, 1996

ISBN ().444.()()()71·2 250 Problems, in Elementary Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathematics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems. Included are problems on several levels of difficulty-some are relatively easy, others rather complex, and a number so abstruse that they originally were the subject of scientific research and their solutions are of comparatively recent date. All of the solutions are given thoroughly and in detail; they contain information on possible generalizations of the given problem and further indicate unsolved problems associated with the given problem and solution.

arXiv (Cornell University), 2016

Following footsteps of Gauss, Euler, Riemann, Hurwitz, Smith, Hardy, Littlewood, Hedlund, Khinchin and Chebyshev, we visit some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We experiment then with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These conjectures relate with Landau or Bunyakovsky or Andrica type conjectures for rational primes. The Landau problem asking whether infinitely many predecessors of primes are square is also related to a determinant problem for the prime matrices under consideration. Some of these matrices are adjacency matrices of bipartite graphs. Their Euler characteristics in turn is related to the prime counting function. When doing statistics of Gaussian primes on rows, we detect a sign of correlations: rows of even distance for example look asymptotically correlated. The expectation values of prime densities were conjectured to converge by Hardy-Littlewood almost 100 years ago. We probe the convergence to these constants, following early experimenters. After factoring out the dihedral symmetry of Gaussian primes, they are bijectively related to the standard primes but the sequence of angles appears random. A similar story happens for Eisenstein primes. Gaussian or Eisenstein primes have now a unique angle attached to them. We also look at the eigenvalue distribution of greatest common divisor matrices whose explicitly known determinants are given number theoretically by Jacobi totient functions and where unexplained spiral patterns can appear in the spectrum. Related are a class of graphs for which the vertex degree density is related to the Euler summatory totient function. We then apply cellular automata maps on prime configurations. Examples are Conway's life and moat-detecting cellular automata which we ran on Gaussian primes. Related to prime twin conjectures and more general pattern conjectures for Gaussian primes is the question whether "life" exists arbitrary far away from the origin, even if is primitive life in form of a blinker obtained from a prime twin. Most questions about Gaussian primes can be asked for Hurwitz primes inside the quaternions, for which the zeta function is just shifted. There is a Goldbach statement for quaternions: we see experimentally that every Lipschitz integer with entries larger than 1 is a sum of two Hurwitz primes with positive entries and every Hurwitz prime with entries larger than 3 is a sum of a Hurwitz and Lipschitz prime. For Eisenstein primes, we see that all but finitely many Eisenstein integers with coordinates larger than 2 can be written as a sum of two Eisenstein primes with positive coordinates. We also predict that every Eisenstein integer is the sum of two Eisenstein primes without any further assumption. For coordinates larger than 1, there are two curious ghost examples. For Octonions, we see that there are arbitrary large Gravesian integer with entries larger than 1 which are not the sum of two Kleinian primes with positive coordinates but we ask whether every Octavian integers larger than some constant K is a sum of two Octavian primes with positive coordinates. Finally we look at some spectra of almost periodic pseudo random matrices defined by Diophantine irrational rotations, where fractal spectral phenomena occur. The matrix is the real part of a van der Monde matrix whose determinant has relations with the curlicue problem in complex analysis or the theory of partitions of integers. Diophantine properties allow to estimate the growth rate of the determinants of these complex matrices if the rotation number is the golden mean.

IOSR Journal of Mathematics, 2012

The present paper studies a new class of numbers. Results obtained in this paper are a table, recurrence relations, generating functions and Summation formulas for these new class of numbers. Many results reduce to their corresponding results for the Catalan numbers .

Advances in Mathematics, 2006

Hardy-Ramanujan Journal, 2019

International audience In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.

2019

We include here some corrections with respect to the published version. 1. We shall make use of this extra information later on. 2. See [29] for a short proof.

In this booklet, I present my proofs of open conjectures on the theory of numbers. It concerns the following conjectures: - The Riemann Hypothesis. - Beal's conjecture. - The conjecture $c<rad^{1.63}(abc)$. - The explicit $abc$ conjecture of Alan Baker. - Two proofs of the $abc$ conjecture. - The conjecture $c<rad^2(abc)$.

In two of my previous published books, “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, respectively “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, I already expressed my passion for integer numbers, especially for primes and Fermat pseudoprimes, fascinating numbers that seem to be a little bit more willing to let themselves ordered and understood than the prime numbers. This book brings together sixty-two papers on prime numbers, many of them supporting the author’s belief, expressed above, namely that new ordered patterns can be discovered in the “undisciplined” set of prime numbers, observing the ordered patterns in the set of Fermat pseudoprimes, especially in the set of Carmichael numbers, the absolute Fermat pseudoprimes, and in the set of Poulet (sometimes also called Sarrus) numbers, the relative Fermat pseudoprimes to base two. Other papers, which are not based on the observation of Fermat pseudoprimes, are based on the observation of Mersenne numbers, Fermat numbers, Smarandache generalized Fermat numbers, and other well known or less known classes of integers which are very much related with the study of primes. Part One of this book of collected papers contains two hundred and thirteen conjectures on primes and Part Two of this book brings together the articles regarding primes, submitted by the author to the preprint scientific database Vixra, representing the context of the conjectures listed in Part One, papers regarding squares of primes, semiprimes, twin primes, sequences of primes, types of duplets or triplets of primes, special classes of composites, ways to write primes, formulas for generating large primes, generalizations of the twin primes and de Polignac’s conjecture, generalizations of Cunningham chains and Fermat numbers and many other classic issues regarding prime numbers. Finally, in the last eight from these collected papers, I defined a new function, the MC function, and I showed some of its possible applications (for instance, I conjectured that for any pair of twin primes p and p + 2, where p ≥ 5, there exist a positive integer n of the form 15 + 18*k such that the value of Smarandache function for n is equal to p and the value of MC function for n is equal to p + 2, I also made a Diophantine analysis of few Smarandache type sequences using the MC function).