Results 21 to 30 of about 147 (145)
Diophantine Equation 41k2−nx+4kny=41k2+nz
Journal of Mathematics, Volume 2026, Issue 1, 2026.Let (a, b, c) be a primitive Pythagorean triple such that a2 + b2 = c2 with 2|b. In 1956, L. Jesmanowicz conjectured that, for any positive integer n, the equation (an)x + (bn)y = (cn)z has only the positive solution (x, y, z) = (2, 2, 2). In 1959, Lu Wenduan claimed that if n = 1 and (a, b, c) = (4k2 − 1, 4k, 4k2 + 1), then the conjecture is true ...
Nai-juan Deng +2 more
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Euclidean algorithms are Gaussian over imaginary quadratic fields
Journal of the London Mathematical Society, Volume 112, Issue 6, December 2025.Abstract We prove that the distribution of the number of steps of the Euclidean algorithm of rationals in imaginary quadratic fields with denominators bounded by N$N$ is asymptotically Gaussian as N$N$ goes to infinity, extending a result by Baladi and Vallée for the real case.
Dohyeong Kim, Jungwon Lee, Seonhee Lim
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Iitaka fibrations and integral points: A family of arbitrarily polarized spherical threefolds
Journal of the London Mathematical Society, Volume 112, Issue 6, December 2025.Abstract Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analog of Iitaka fibrations, which have not yet been adequately
Ulrich Derenthal, Florian Wilsch
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A circle method approach to K‐multimagic squares
Journal of the London Mathematical Society, Volume 112, Issue 3, September 2025.Abstract In this paper, we investigate K$K$‐multimagic squares of order N$N$. These are N×N$N \times N$ magic squares that remain magic after raising each element to the k$k$th power for all 2⩽k⩽K$2 \leqslant k \leqslant K$. Given K⩾2$K \geqslant 2$, we consider the problem of establishing the smallest integer N2(K)$N_2(K)$ for which there exist ...
Daniel Flores
wiley +1 more source
On a conjecture on exponential Diophantine equations [PDF]
Acta Arithmetica, 2009 We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We
Cipu, Mihai, Mignotte, Maurice
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Computer Applications in Engineering Education, Volume 33, Issue 4, July 2025.ABSTRACT Integer and modular arithmetic is a fundamental area of mathematics, with extensive applications in computer science, and is essential for cryptographic protocols, error correction, and algorithm efficiency. However, students often struggle to understand its abstract nature, especially when transitioning from theoretical knowledge to practical
Violeta Migallón +2 more
wiley +1 more source
Jeśmanowicz' conjecture on exponential diophantine equations
Functiones et Approximatio Commentarii Mathematici, 2011 Jeśmanowicz' conjecture is the following statement: If \(a\), \(b\), \(c\) are coprime positive integers such that \(a^2+b^2=c^2\) with even \(b\), then the exponential equation \(a^x+b^y=c^z\) has the only solution \((x,y,z)=(2,2,2)\) in positive integers. This paper contains various new results on this conjecture.
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Localized and Extended Phases in Square Moiré Patterns
Annalen der Physik, Volume 537, Issue 6, June 2025.Rotated superimposed lattices in two dimensions, the termed moiré patterns, represent a clear example of how the structure affects the physical properties of a particle moving on it. A robust numerical treatment of continuous and discrete models leads to confirm that while localized states result from angles that produce non‐commensurable lattices ...
C. Madroñero +2 more
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On an Erdős similarity problem in the large
Bulletin of the London Mathematical Society, Volume 57, Issue 6, Page 1801-1818, June 2025.Abstract In a recent paper, Kolountzakis and Papageorgiou ask if for every ε∈(0,1]$\epsilon \in (0,1]$, there exists a set S⊆R$S \subseteq \mathbb {R}$ such that |S∩I|⩾1−ε$\vert S \cap I\vert \geqslant 1 - \epsilon$ for every interval I⊂R$I \subset \mathbb {R}$ with unit length, but that does not contain any affine copy of a given increasing sequence ...
Xiang Gao +2 more
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Exponential diophantine equations with four terms
Indagationes Mathematicae, 1992 This article gives some examples how to make exponential diophantine equations more practical. The authors take the large exponential bounds for solutions given by Baker's method to computational available bounds. Let \(p\) and \(q\) be distinct primes less than 200. The main theorems are: (1) Every solution of the equation \(p^ x q^ y\pm p^ z \pm q^ w
Deze, Mo, Tijdeman, R.
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