Results 41 to 50 of about 3,148,152 (262)

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

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 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

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

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
core   +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),
openaire   +1 more source

Some bounds related to the 2‐adic Littlewood conjecture

Mathematika, Volume 72, Issue 2, April 2026.
Abstract For every irrational real α$\alpha$, let M(α)=supn⩾1an(α)$M(\alpha) = \sup _{n\geqslant 1} a_n(\alpha)$ denote the largest partial quotient in its continued fraction expansion (or ∞$\infty$, if unbounded). The 2‐adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational α$\alpha$ such that M(2kα)$M(2^k \alpha)$ is ...
Dinis Vitorino, Ingrid Vukusic
wiley   +1 more source

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

A Method to Solve the Diophantine Equation $ax^2-by^2+c=0$

, 2006
It is a generalization of Pell's equation $x^2-Dy^2=0$. Here, we show that: if our Diophantine equation has a particular integer solution and $ab$ is not a perfect square, then the equation has an infinite number of solutions; in this case we find a ...
Smarandache, Florentin
core   +1 more source

Plank theorems and their applications: A survey

Bulletin of the London Mathematical Society, Volume 58, Issue 1, January 2026.
Abstract Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.
William Verreault
wiley   +1 more source