On the Diophantine equation 1k + 2k + ⋯ + xk = yn

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.

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).

Journal of Number Theory, 2011

The title equation, where p > 3 is a prime number ≡ 7 (mod 8), q is an odd prime number and x, y, n are positive integers with x, y relatively prime, is studied. When p ≡ 3 (mod 8), we prove (Theorem 2.3) that there are no solutions. For p ≡ 3 (mod 8) the treatment of the equation turns out to be a difficult task. We focus our attention to p = 5, by reason of an article by F.

Archiv der Mathematik, 2008

Let 2 ≤ p < 100 be a rational prime and consider equation in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in gcd(x, y) = 1, our Theorem is an extension of several earlier results , [

2010

In this note, we resolve the Diophantine equation x 2 + 2 a · 11 b = y n with coprime positive integers x, y and positive integers n ≥ 3.

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.

In this note we present a method of solving this Diophantine equation, method which is different from Ljunggren’s, Mordell’s, and R.K.Guy’s.

International Journal of Innovative Technology and Exploring Engineering, 2019

In this article, we prove that the non-linear Diophantine equation = 2 1 2 … + 1; ≥ 2, ∈ − {2}, ′ are distinct and P is the set of all prime numbers has an infinite number of solutions using the notion of a periodic sequence. Then we also obtained certain results concerning Euler Mullin sequence.

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.

2010