Results 61 to 70 of about 1,430,683 (165)
Multiplicatively dependent integer vectors on a hyperplane
Journal of the London Mathematical Society, Volume 114, Issue 1, July 2026.Abstract We establish several asymptotic formulae and upper bounds for the count of multiplicatively dependent integer vectors that lie on a fixed affine hyperplane and have bounded height. This work constitutes a direct extension of the results obtained by Pappalardi, Sha, Shparlinski, and Stewart.
Muhammad Afifurrahman +2 more
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The Polynomial Solutions of Quadratic Diophantine Equation X2−ptY2+2KtX+2ptLtY = 0
Journal of Mathematics, 2021 In this study, we consider the number of polynomial solutions of the Pell equation x2−pty2=2 is formulated for a nonsquare polynomial pt using the polynomial solutions of the Pell equation x2−pty2=1.
Hasan Sankari, Ahmad Abdo
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Definability of complex functions in o‐minimal structures
Journal of the London Mathematical Society, Volume 114, Issue 1, July 2026.Abstract We prove that holomorphic continuations of functions in the classes an∗$\mathbf {an}^*$ and G$\mathcal {G}$ are definable in the o‐minimal structures Ran∗$\mathbb {R}_{\operatorname{an}^*}$ and RG$\mathbb {R}_{\mathcal {G}}$, respectively. More specifically, we give complex domains on which the holomorphic continuations are definable and show ...
Adele Padgett, Patrick Speissegger
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A note on the ternary Diophantine equation x2 − y2m = zn
Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2021 Let ℕ be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x2 − y2m = zn, x, y, z, m, n ∈ ℕ, gcd(x, y) = 1, m ≥ 2, n ≥ 3.
Bérczes Attila +3 more
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Random Diophantine equations in the primes
Mathematika, Volume 72, Issue 3, July 2026.Abstract We consider equations of the form a1x1k+⋯+asxsk=0$a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}=0$ where the variables xi$x_{i}$ are all taken to be primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove that, whenever k⩾2$k\geqslant 2$, s⩾3k+2$s\geqslant 3k+2$, this holds
Philippa Holdridge
wiley +1 more source
On the Diophantine equation x2+p2k+1=4yn
International Journal of Mathematics and Mathematical Sciences, 2002 It has been proved that if p is an odd prime, y>1, k≥0, n is an integer greater than or equal to 4, (n,3h)=1 where h is the class number of the field Q(−p), then the equation x2+p2k+1=4yn has exactly five families of solution in the positive integers x ...
S. Akhtar Arif, Amal S. Al-Ali
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, 1967 Formulae are given furnishing all non-trivial integer solutions of the equation \[ (x^2-t^2)(y^2-t^2)=\biggl(\biggl({y-x\over 2}\biggr)^2-t^2\biggr)^2 \] considered for \(t=1\) by the reviewer and \textit{W. Sierpiński} [Elem. Math. 18, 132--133 (1963; Zbl 0126.07301)].
openaire +2 more sources
A Repulsion Motif in Diophantine Equations [PDF]
The American Mathematical Monthly, 2011 Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has only finitely many integral solutions.
Everest, G, Ward, T
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On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0 [PDF]
, 2003 summary:In this study, we determine when the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\leq n\leq 10.$ Moreover, we give all positive integer solutions of the same equation for ...
Keskin, Refik +9 more
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New results on embeddings of self‐similar sets via renormalization
Proceedings of the London Mathematical Society, Volume 133, Issue 1, July 2026.Abstract For self‐similar sets X,Y⊆R$X,Y\subseteq \mathbb {R}$, we obtain new results toward the affine embeddings conjecture of Feng–Huang–Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of X,Y$X,Y$ have algebraic contraction ratios, and also for arbitrary Y$Y$ when the maps ...
Amir Algom, Michael Hochman, Meng Wu
wiley +1 more source