Results 41 to 50 of about 20,685 (194)

Random Diophantine equations, I

Advances in Mathematics, 2014
Comment: The results in this paper use an $L^2$-technique and supersede those in an earlier version (see arXiv:1110.3496) that relied on an $L^1$-argument, but for instructional purposes we found it useful to keep the older, technically simpler version.
Brüdern, Jörg, Dietmann, Rainer
openaire   +4 more sources

On a Diophantine Equation [PDF]

Journal of the London Mathematical Society, 1951
Summary: Denote by \(N(a,b)\) the smallest integer \(n\) so that \[ \frac{a}{b}=\frac{1}{x_1}+\cdots+\frac{1}{x_n},\quad ...
openaire   +4 more sources

The diophantine equation r2+r(x+y)=kxy

International Journal of Mathematics and Mathematical Sciences, 1985
The Diophantine equation of the title is solved in integers.
W. R. Utz
doaj   +1 more source

Diophantine equations involving factorials [PDF]

Mathematica Bohemica, 2017
We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only
Horst Alzer, Florian Luca
doaj   +1 more source

On primitive integer solutions of the Diophantine equation $t^2=G(x,y,z)$ and related results

, 2015
In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if $F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,
Gawron, Maciej, Ulas, Maciej
core   +1 more source

Studies of Positive Integer Solutions of the Diophantine Equation x2−ay2−bx−cy−d=0 by the Transformation Method

Journal of Mathematics
Solving the Diophantine equation has fascinated mathematicians from various civilizations. In this paper, we propose the resolution of quadratic Diophantine equations with integer coefficients.
Francklin Fenolahy, Harrimann Ramanantsoa, André Totohasina   +2 more
doaj   +1 more source

Three Diophantine equations concerning the polygonal numbers [PDF]

Notes on Number Theory and Discrete Mathematics
Many authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation mPₖ(x)+nPₖ(y)=z², where Pₖ(x) denotes the x-th k-polygonal number and m, n are positive integers.
Yong Zhang, Mei Jiang, Qiongzhi Tang
doaj   +1 more source

Systems of Diophantine Equations [PDF]

Proceedings of the American Mathematical Society, 1951
where fi and gi are homogeneous polynomials with integral coefficients, fi being of degree n and gi being of degree m. If there are no integers s> 1, a k, 3' such that ak = sla , ij = s, where X, g are positive integers such that Xn =,m, then Xk= ak, yij=gi3 is defined to be a primitive solution of (1). If Xk=aQk, yij=fi3 is a primitive solution of (1),
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Moderate Deviation Principles for Lacunary Trigonometric Sums

Mathematische Nachrichten, Volume 299, Issue 5, Page 1028-1044, May 2026.
ABSTRACT Classical works of Kac, Salem, and Zygmund, and Erdős and Gál have shown that lacunary trigonometric sums despite their dependency structure behave in various ways like sums of independent and identically distributed random variables. For instance, they satisfy a central limit theorem (CLT) and a law of the iterated logarithm.
Joscha Prochno, Marta Strzelecka
wiley   +1 more source

More Than 1700 Years of Word Equations

, 2015
Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the
A Boudet, A Kościelski, J Ferté, L Ciobanu, O Kharlampovich, PR Asveld, RM Karp, V Diekert, W Plandowski, W Plandowski, Z Sela   +10 more
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