On Systems of Linear Diophantine Equations

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2020

This is a redaction of the Inaugural Lecture the author gave at the University of Hyderabad in January 2019 in honor of the late great Geometer (and Fields medalist) Maryam Mirzakhani. What is presented here is a limited perspective on a huge field, a meandering path through a lush garden, ending with a circle of problems of current interest to the author. No pretension (at all) is made of being exhaustive or current. 1. Something light to begin with When Nasruddin Hodja claimed that he could see in the dark, his friend pointed out the incongruity when Hodja was seen carrying a lit candle at night. ”Not so,” said Nasruddin, ”the role of the light is for others to be able to see me.” The moral is of course that one needs to analyze all possibilities before asserting a conclusion. Maryam Mirzakhani, whom this Lecture is named after, would have liked the stories of Hodja. Mirzakhani’s mathematical work gave deep insights into the structure of geodesic curves on hyperbolic surfaces. Suc...

Proceedings Mathematical Sciences

To any Diophantine equation with integral coefficients we associate a finitely generated abelian group. The analysis of this group by row-reduction generally leads to simpler equations which are equivalent to the original but often dramatically easier to solve. This method of studying equations is useful over finite fields as well as over Q. Some applications and an example are discussed.

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.

Universal Journal of Mathematics and Applications, 2020

The paper considers a linear Diophantine equation. A method (algorithm) for finding a general class of solutions of equation is proposed. The proposed algorithm is explained by examples of equations with two and three variables, trying to direct the reader to a general idea that describes the essence of the method used.

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.

Journal of Number Theory, 2005

Let D > 2 be a positive integer, and let p be an odd prime not dividing D. In this paper, using the deep result of Bilu, Hanrot and Voutier (i.e., the existence of primitive prime factors of Lucas and Lehmer sequences), by computing Jacobi's symbols and using elementary arguments, we prove that: if (D, p) = (4, 5), (2, 5), then the diophantine equation x 2 + D m = p n has at most two positive integer solutions (x, m, n). Moreover, both x 2 + 4 m = 5 n and x 2 + 2 m = 5 n have exactly three positive integer solutions (x, m, n).

Definitions and Properties of the Integer Solution of a Linear System.

Computer Algebra in Scientific Computing CASC 2001, 2001

2000

This paper has been updated and completed thanks to suggestions and critics coming from Dr. Mike Hirschhorn, from the University of New South Walles. We want to express our highest gratitude. The paper appeared in an abbreviated form (6). The present work is a complete form. For the homogeneous diophantine equations: x 2 + by 2 + cz 2 =