On the Diophantine Equation

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.

arXiv (Cornell University), 2019

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f (x) ∈ Z[x] monic and q 1 , • • • , qm ∈ Z, we study the conditions for which the Diophantine equation has finitely many solutions in integers. Also assuming ABC-Conjecture, we study the conditions for finiteness of integer solutions of the Diophantine equation f (x) = g(y).

Indagationes Mathematicae, 2003

Journal of Number Theory, 2016

In this paper we investaigate Diophantine equations of the form T 2 = G(X), X = (X 1 ,. .. , Xm), where mainly m = 3 or m = 4 and G specific homogenous quintic form. First, we prove that if F (x, y, z) = x 2 + y 2 + az 2 + bxy + cyz + dxz ∈ Z[x, y, z] and (b − 2, 4a − d 2 , d) = (0, 0, 0), then the Diophantine equation t 2 = nxyzF (x, y, z) has solution in polynomials x, y, z, t with integer coefficients, without polynomial common factor of positive degree. In case a = d = 0, b = 2 we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideration. As an application of our result we prove that for each n ∈ Q \ {0} the Diophantine equation 2010 Mathematics Subject Classification. 11D41.

american mathematical society, 1990

Abstract. Let f(x),g(y) be polynomials over Z of degrees n and m re-spectively and with leading coefficients an , b . Suppose that m\n and that an/bm is the mth power of a rational number. We give two elementary proofs that the equation f(x) = g(y) has at most finitely many integral ...

Journal of Number Theory, 2014

This note presents corrections to the paper by Y. Wang and T. Wang [2]. The unique theorem given in that paper states that for any odd integer n > 1, nx 2 + 2 2m = y n has no positive integer solution (x, y, m) with gcd(x, y) = 1.

2003

Given r> 2, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer co- ecients mapping the \cube" with real coordinates from ( r; r )i nto ( t;t). This directly translates to a nice statement in logic (more specically recursion theory) with a correspond- ing phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable. In some private communications, Harvey Friedman raised the following problem: given r> 2, give an upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coecients mapping the \cube" with real variables from ( r; r )i nto ( t; t). Robin Peman...

2010

Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Combinatorial, arithmetical, algebraic, topological and dynamical aspects CIRM International center of mathematics meetings, 163 avenue de luminy 13288 MARSEILLE Organizing committee Sabine EVRARD AMIENS Youssef FARES AMIENS Amandine LERICHE AMIENS Jean-Luc CHABERT AMIENS Paul-Jean CAHEN MARSEILLE Scientific committee Paul-Jean CAHEN FRANCE Jean-Luc CHABERT FRANCE Stefania GABELLI ITALY Byung KANG SOUTH COREA Roger WIEGAND USA MONDAY THUESDAY WEDNESDAY THURSDAY FRIDAY 9H 9H/9H30 9H/9H30 9H40/10H 9H40/10H 9H50/10H30 9H50/10H30 Alice FABBRI Valentina BARUCCI 10H 10H10/10H30 10H10/10H30 Said El BAGHDADI Marco FONTANA 11H 11H/11H40 11H/11H30 11H/11H30 11H/11H30 11H/11H30 11H40/12H 11H40/12H10 11H40/12H 11H40/12H 11H50/12H30 Vadim PONOMARENKO Gabriele FUSACCHIA Gabriel PICAVET 12H 12H10/12H30 12H10/12H30 12H10/12H30 Faten KHOUJA Driss KARIM Amor HAOUAOUI 14H/14H20 Mohamed KHALIFA 14H30 14H30/15H...

Contributions to the Theory of Transcendental Numbers, 1984

Diophantine arithmetic is one of the oldest branches of mathematics, the search for integer or rational solutions of algebraic equations. Pythagorean triangles are an early instance. Diophantus of Alexandria wrote the first related treatise in the fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat. The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations y 2 = x 6 + k, k = −39, −47, the two previously unsolved cases for |k| < 50, are solved using algebraic number theory and the elliptic Chabauty method. The thesis also studies the genus three quartic curves F (x 2 , y 2 , z 2) = 0 where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals. The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form n = (x+y +z +w)(1/x+1/y +1/z +1/w). Further, an example, the first such known, of a quartic surface x 4 + 7y 4 = 14z 4 + 18w 4 is given with remarkable properties: it is everywhere locally solvable, yet has no nonzero rational point, despite having a point in (non-trivial) odd-degree extension fields i of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves. ii ACKNOWLEDGEMENTS I would like to thank my advisor Professor Andrew Bremner for his guidance, his generosity, his encouragement and his kindness during my graduate years. Without his help and support, I will not be able to finish the thesis. I show my most respect to him, both his personality and his mathematical expertise. I would like to thank Professor Susanna Fishel for some talks we had. These talks did encourage me a lot at the beginning of my graduate years. I would like to thank other members of my Phd committee, Professor John Quigg, Professor John Jones, and Professor Nancy Childress. I would like to thank the school of mathematics and statistical sciences at Arizona State University for all the funding and support. And finally, I would like to thank the members in my family. My grandmother, my father, my mom, Mr Phuong and his wife Mrs Doi and their son Phi, and to my cousin Mr Tan for all of their constant support and encouragement during my undergraduate and my graduate years.

Acta Arithmetica, 2003

Among other things we show that for each n-tuple of positive rational numbers (a 1 ,. .. , a n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 +• • •+a n x n = 1 with x 1 ,. .. , x n S-units are not contained in fewer than exp((4 + o(1))s 1/2 (log s) −1/2) proper linear subspaces of C n. This generalizes a result of Erdős, Stewart and Tijdeman [7] for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 ≤ m < n, and any sufficiently large s there are integers α 0 ,. .. , α m in K which are linearly independent over Q, and prime numbers p 1 ,. .. , p s , such that the norm polynomial equation |N K/Q (α 0 + α 1 x 1 + • • • + α m x m)| = p z 1 1 • • • p zs s has at least exp{(1+o(1)) n m s m/n (log s) −1+m/n } solutions in x 1 ,. .. , x m , z 1 ,. .. , z s ∈ Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an effective lower bound for the number ψ K,T (X, Y) of ideals in a number field K of norm ≤ X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm ≤ Y. This generalizes results of Canfield, Erdős and Pomerance [6] and of Moree and Stewart [19].