Results 161 to 170 of about 81,722 (194)

Entire solutions of Fermat type differential-difference equations

Archiv der Mathematik, 2012
Equations of the form \[ f(z)^n+f(z+c)^m=1,\, f'(z)^n+f(z+c)^m=1 \] and \[ f(z)^n+(f(z+c)-f(z))^m=1 \] are called Fermat-type differential-difference equations, where \(m, n\) are positive integers. The authors prove that some transcendental entire solutions of finite order to these equations, if any, can be expressed by sine functions.
Liu, Kai, Cao, Tingbin, Cao, Hongzhe
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On Entire Solutions of Two Certain Fermat-Type Differential–Difference Equations

Bulletin of the Malaysian Mathematical Sciences Society, 2019
The entire solutions of the Fermat-type differential-difference equation \[ F^{n}+m \omega FG+G^{n}=e^{\alpha z+ \beta}, \;\; m,n \in \mathbb{Z}, \;\; m,n>0, \;\; \omega, \alpha, \beta \in \mathbb{C}, \;\; \omega \notin \{ 0,1 \} \] are investigated. The authors use Nevanlinna theory for proving their results.
Wang, Qiong, Chen, Wei, Hu, Peichu
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On Entire Solutions of Two Fermat-Type Differential-Difference Equations

Bulletin of the Iranian Mathematical Society
The authors investigate the equation \[ \begin{multlined} [a_0f(z)+a_1f^{\prime\prime}(z)+a_2f(z+c)]^2+2\omega [a_0f(z)+a_1f^{\prime\prime}(z)+a_2f(z+c)]\cdot \\ [b_0f(z)+b_1f^{\prime\prime}(z)+b_2f(z+c)]+[b_0f(z)+b_1f^{\prime\prime}(z)+b_2f(z+c)]^2=e^{\alpha z+\beta}, \end{multlined} \] where \(\alpha, \beta, \omega^2(\neq 0,1), c(\neq 0), a_i, b_i (i=
Yihui Gong, Qi Yang
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Entire and meromorphic solutions of fermat type partial differential equations

Analysis, 1999
The author studies complex analytic solutions of PDEs of the form \({\mathfrak E}_{n,m}u:= \sum^n_{i= 1}(u_{z_i})^m= 1\), where \(u: \mathbb{C}^n\to \mathbb{C}\) (or \(\widehat{\mathbb{C}}\)), \(z_i\in \mathbb{C}\), \(n,m\in\mathbb{N}\). He uses and refines results from the better known situation where the \(u_{z_i}\) are replaced by general analytic ...
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Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations

Mediterranean Journal of Mathematics, 2018
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ling Xu, Tingbin Cao
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Arithmetic of cyclotomic fields and Fermat-type equations

1994
Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the
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On Fermat-Type Functional and Partial Differential Equations

2012
This paper concerns entire and meromorphic solutions to functional and nonlinear partial differential equations of the form a 1 f m +a 2 g n =a 3 with function coefficients a k , k=1,2,3, where f and g are unknown functions or partial derivatives of an unknown function.
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Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Analysis and Mathematical Physics
The topic of the paper is complex functional equations of the form \[ f(z)^n+g(z)^n+h(z)^n+k(z)^n=1, \] which are referred to as the bi-Fermat type equations. The results presented consider two particular types of such equations: quadratic partial differential equations, \[ f(z_1,z_2)^2+\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2+g(z_1,z_2)^
Gao, Yingchun, Liu, Kai
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Local criteria for the unit equation and the asymptotic Fermat’s Last Theorem

Proceedings of the National Academy of Sciences of the United States of America, 2021
Nuno Freitas, Alain Kraus, Samir Siksek   +2 more
exaly  

On Ternary Equations of Fermat Type and Relations with Elliptic Curves

1997
The main purpose of this chapter is to show how arithmetical properties of elliptic curves E defined over global fields K and corresponding Galois representations are often related to interesting diophantine questions, amongst which the most prominent is without doubt Fermat’s Last Theorem, which has now become Wiles’ theorem.
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