Representation of Integers by Near Quadratic Sequences

Journal of Number Theory, 2003

We give equivalent formulations of the Erd + os-Tura´n conjecture on the unboundedness of the number of representations of the natural numbers by additive bases of order two of N: These formulations allow for a quantitative exploration of the conjecture. They are expressed through some functions of xAN reflecting the behavior of bases up to x: We examine some properties of these functions and give numerical results showing that the maximum number of representations by any basis is X6:

In this paper, we present eighteen interesting infinite products and their Lambert series expansions. From these, we deduce formulae for the number of representations of an integer n by eighteen quadratic forms in terms of divisor sums. -Dedicated to the memory of my grandmother Yuet Kwai Mah.

Math Notes Engl Tr, 2008

In this paper, we use the following standard notation: Z is the ring of integers, Q and Q are the fields of rational and algebraic numbers, respectively, ϕ(q) is the Euler function, and ω(q) is the number of different prime divisors of a number q.

2000

Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number. Let q be a rational number. A set of non-zero rationals {a1, a2,. .. , am} is called a rational Diophantine m-tuple with the property D(q) if aiaj + q is a square of a rational number for all 1 ≤ i < j ≤ m. It is easy to prove that for every rational number q there exist infinitely many distinct rational Diophantine quadruples with the property D(q). Thus we come to the following open question: For which rational numbers q there exist infinitely many distinct rational Diophantine quintuples with the property D(q)? In the present paper we give an affirmative answer to the above question for all rationals of the forms q = r 2 and q = −3r 2 , r ∈ Q.

Journal of Number Theory, 2002

Mathematics and Statistics, 2021

Although it is true that there are several articles that study quadratic equations in two variables, they do so in a general way. We focus on the study of natural numbers ending in one, because the other cases can be studied in a similar way. We have given the subject a different approach, that is why our bibliographic citations are few. In this work, using basic tools of functional analysis, we achieve some results in the study of integer solutions of quadratic polynomials in two variables that represent a given natural number. To determine if a natural number ending in one is prime, we must solve equations (i) P = (10x + 9)(10y + 9), (ii) P = (10x + 1)(10y + 1), (iii) P = (10x + 7)(10y + 3). If these equations do not have an integer solution, then the number P is prime. The advantage of this technique is that, to determine if a natural number p is prime, it is not necessary to know the prime numbers less than or equal to the square root of p. The objective of this work was to reduce the number of possibilities assumed by the integer variables (x, y) in the equation (i), (ii), (iii) respectively. Although it is true that this objective was achieved, we believe that the lower limits for the sums of the solutions of equations (i), (ii), (iii), were not optimal, since in our recent research we have managed to obtain limits lower, which reduce the domain of the integer variables (x, y) solve equations (i), (ii), (iii), respectively. In a future article we will show the results obtained. The methodology used was deductive and inductive. We would have liked to have a supercomputer, to build or determine prime numbers of many millions of digits, but this is not possible, since we do not have the support of our respective authorities. We believe that the contribution of this work to number theory is the creation of linear functionals for the study of integer solutions of quadratic polynomials in two variables, which represent a given natural number. The utility of large prime numbers can be used to encode any type of information safely, and the scheme shown in this article could be useful for this process.

Finite Fields and Their Applications, 2010

In this work we obtain a nontrivial estimate for the size of the set of triples (a, b, c) 2 F ⇤ q ⇥ F q ⇥ F q which correspond to stable quadratic polynomials f (x) = aX 2 + bX + c over the finite field F q with q odd. This estimate is an step towards the bound O(q 11/4 ) conjectured in a recent work of A. Ostafe and I. Shparlinski.

Journal of Number Theory, 2002

Mathematical Proceedings of the Cambridge …, 2002

arXiv (Cornell University), 2007

Discrete Mathematics, 1987

Periodica Mathematica Hungarica, 2006

The periodicity of sequences of integers (an) n∈Z satisfying the inequalities 0 ≤ a n−1 + λan + a n+1 < 1 (n ∈ Z)

In three of my previous published books, namely “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function” and “Two hundred and thirteen conjectures on primes”, I showed my passion for conjectures on sequences of integers. In spite the fact that some mathematicians stubbornly understand mathematics as being just the science of solving and proving, my books of conjectures have been well received by many enthusiasts of elementary number theory, which gave me confidence to continue in this direction. Part One of this book brings together papers regarding conjectures on primes, twin primes, squares of primes, semiprimes, different types of pairs or triplets of primes, recurrent sequences, sequences of integers created through concatenation and other sequences of integers related to primes. Part Two of this book brings together several articles which present the notions of c-primes, m-primes, c-composites and m-composites (c/m-integers), also the notions of g-primes, s-primes, g-composites and s-composites (g/s-integers) and show some of the applications of these notions (because this is not a book structured unitary from the beginning but a book of collected papers, I defined the notions mentioned in various papers, but the best definition of them can be found in Addenda to the paper numbered tweny-nine), in the study of the squares of primes, Fermat pseudoprimes and generally in Diophantine analysis. Part Three of this book presents the notions of “Coman constants” and “Smarandache-Coman constants”, useful to highlight the periodicity of some infinite sequences of positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers), respectively in the analysis of Smarandache concatenated sequences. Part Four of this book presents the notion of Smarandache-Coman sequences, id est sequences of primes formed through different arithmetical operations on the terms of Smarandache concatenated sequences. Part Five of this book presents the notion of Smarandache-Coman function, a function based on the well known Smarandache function which seems to be particularly interesting: beside other characteristics, it seems to have as values all the prime numbers and, more than that, they seem to appear, leaving aside the non-prime values, in natural order.

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.

Bulletin of the London Mathematical Society, 2010

|A| ≥ αX and |B| ≥ βX, we show that the number of rational numbers expressible as a/b with (a, b) in A × B is ≫ (αβ) 1+ǫ XY for any ǫ > 0, where the implied constant depends on ǫ alone. We then construct examples that show that this bound cannot in general be improved to ≫ αβXY. We also resolve the natural generalisation of our problem to arbitrary subsets C of the integer points in [1, X] × [1, Y ]. Finally, we apply our results to answer a question of Sárközy concerning the differences of consecutive terms of the product sequence of a given integer sequence.