On a problem of Diophantus for rationals

In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.

Journal of Number Theory, 2004

In this paper, we prove that if {a, b, c, d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then (a + b − c − d) 2 = 4(ab + 1)(cd + 1). This settles the "strong" Diophantine quintuple conjecture for polynomials with integer coefficients.

The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the product of the reversals of $A$ and of $B$. A pair $(A,B)$ of positive integers is said to be polynomial if the product $A \times B$ can be performed without carry. In this paper, polynomial pairs are introduced to study palindromic pairs. While polynomial pairs are readily seen to be palindromic, the converse does not hold in general but is conjectured to be true when neither $A$ nor $B$ is a palindrome. Connection is made with classical topics in recreational mathematics such as reversal multiplication, palindromic squares, repunits, and Lychrel numbers.

Rocky Mountain Journal of Mathematics, 2007

Let m ≥ 2 and k ≥ 2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a 1 , a 2 , . . . , a m ) of non-zero elements of R such that a i a j + 1 is a k-th power of an element of R for 1 ≤ i < j ≤ m. In this paper, we investigate the case when k ≥ 3 and R = K[X], the ring of polynomials with coefficients in a field K of characteristic zero. We prove the following upper bounds on m, the size of diophantine m-tuple: m ≤ 5 if k = 3; m ≤ 4 if k = 4; m ≤ 3 for k ≥ 5; m ≤ 2 for k even and k ≥ 8.

Rocky Mountain Journal of Mathematics, 2003

In this paper, we prove that there does not exist a set of four polynomials with integer coefficients, which are not all constant, such that the product of any two of them is one greater than a square of a polynomial with integer coefficients.

Fibonacci Quart., to appear

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.

International journal of number theory, 2008

In this paper, we prove that there does not exist a set of 11 polynomials with coefficients in a field of characteristic 0 with the property that the product of any two distinct elements plus 1 is a perfect square. Moreover, we prove that there does not exist a set of 5 polynomials and the property that the product of any two distinct elements plus 1 is a perfect kth power with k ≥ 7. Combining these results, we get an absolute upper bound for the size of a set with the property that the product of any two elements plus 1 is a pure power.

Mathematical Proceedings of the Cambridge Philosophical Society

We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽 q [t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.

2014

This paper was first completed on July24, 2008. It was first published in the Journal MATHEMATICAL SPECTRUM, on September 2010; specifically in the 2010/2011 issue, No1, Vol. 43; on page 9. In David Burton's book " The History of Mathematics, see reference[1]; the following problem can be found: "Find three (rational) numbers such that the sum of the product of any two of them, added to the square of the third one equals the square of a rational number," It is listed as an exercise in the above book, and it can be found in Book III, problem14 of Diophantus' Arithmetica( see historical note in Section6). We discuss Diophantus' approach in how to solve the above problem; and how to generalize it. We do so on pages 1 and 2. And this bring us to the following key definition, Definition1, on page3: Definition1: Let L(x) , M(x) be degree one polynomials with integer coefficients and c a nonzero integer, We say that the ordered triple (L(x) , M(x) , c ) has the property P if there exist linear polynomials l(x) and m(x) , with integer coefficients, and such that; L(x) M(x) + c^2 = [l(x)]^ , and [L(x)]^2 + c M(x) = [m(x)]^2. The work we do on pages 4 and 5 ; allows to determine all the above described ordered triples that have property P; see box below equation (10) on page 5. After that, in the second half of page5, we solve the generalized version of Diophantus' problem .