Fifteen problems in number theory

Mathematics and Statistics, 2022

In this work we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or P = (10x + 7)(10y + 3) where P is a given natural number that ends in one. This allows us to decide the primality of a natural number P that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension n, n ≥ 2 , in which the relation is obtained: a 1 + a 2 + • • • + a n = λ[a 2 1 + • • • + a 2 n ], where a i is a real number for i = 1, ..., n. When n = 3 or n = 2 we manage to address Fermat's Last Theorem and the equation x 4 + y 4 = z 4 , proving that both equations do not have positive integer solutions. For n = 2, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.

In this paper, we will discuss two questions, the primality test for Fermat number Fn and several fundamental problems in transfinite number theory. For the primality test for Fermat number Fn, this paper introduces the method of primality test for Fermat number Fn and the Fermat factorization theorem. Meanwhile, the factorization theorem is used to write operation procedure, so that we can distinguish whether 2n+2t+1 is the divisor of Fn or not. For some of the fundamental problems in transfinite number theory, this paper introduces the concept of countable expansion, and then inspects countability of the set of real numbers from the aspect of expansion. Besides, we analyze the cardinal number of linear point set, deriving the fact that their cardinal number possesses invariance.

The goal of these several lectures is to discuss in more details some properties of integers. In what follows Z = {0, 1, −1, 2, −2, 3, −3,. .. , n, −n,. .. } will denote the set of all integers and it will be our universe of discourse. By N = {1, 2, 3,. .. , n,. . .} we denote the set of positive integers. If otherwise is stated, letters a, b, c,. .. , x, y, z will be used to represent integers only, and we will often allow ourselves not to mention this in the future. We do not give a formal definition of integers, and assume that the reader is well familiar with their basic properties, such as: We also assume that the First and the Second Principles of Mathematical Induction are valid methods of proving statements of the form ∀n ∈ Z ≥n0 [P (n)], where Z ≥n0 is the set of all integers greater or equal to an integer n 0. We will also use The Well-Ordering Axiom for Z ≥n0 : Every non-empty subset of Z ≥n0 contains unique smallest element. It can be shown that the Well-Ordering Axiom is equivalent to each of the two Principles of Mathematical Induction. By |n| we denote the absolute value of n, which is equal to n if n ≥ 0, and is −n if n < 0 (e.g. |5| = 5, |0| = 0, | − 7| = −(−7) = 7). For every two integers a, b, |ab| = |a| • |b|.

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.

Bolyai Society Mathematical Studies, 2013

1. We shall make use of this extra information later on. 2. See [29] for a short proof.