Relevant bibliographies by topics / Generalized Fermat equation

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Generalized Fermat equation'

Author: Grafiati
Published: 3 June 2025
Last updated: 20 July 2025

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Journal articles on the topic "Generalized Fermat equation"

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Bennett, Michael A., Imin Chen, Sander R. Dahmen, and Soroosh Yazdani. "Generalized Fermat equations: A miscellany." International Journal of Number Theory 11, no. 01 (2014): 1–28. http://dx.doi.org/10.1142/s179304211530001x.

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This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all p
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DĄBROWSKI, ANDRZEJ. "ON A CLASS OF GENERALIZED FERMAT EQUATIONS." Bulletin of the Australian Mathematical Society 82, no. 3 (2010): 505–10. http://dx.doi.org/10.1017/s000497271000033x.

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AbstractWe generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation xn+yn=2αpz2’, C. R. Math. Acad. Sci. Soc. R. Can.28 (2006), 6–11] concerning the solubility of the diophantine equation xn+yn=2αpz2. We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3,p) or (4,2,p) can be reduced, in certain cases, to studying several equations of the type (p,p,2).
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Dr., N. Thiruniraiselvi*1 &. Dr. M.A. Gopalan2. "OBSERVATIONS ON." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 5, no. 10 (2018): 162–68. https://doi.org/10.5281/zenodo.1475189.

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This paper aims at determining the non-zero distinct integer solutions to the Fermat equation of the form . In particular, we have presented integer solutions to two different choices of Fermat equations represented by  &  . A few interesting relations between the solutions are also given.
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Chen, Imin. "A Diophantine Equation Associated to X0(5)." LMS Journal of Computation and Mathematics 8 (2005): 116–21. http://dx.doi.org/10.1112/s1461157000000929.

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AbstractSeveral classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat e
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Bombieri, E., and J. Mueller. "The generalized fermat equation in function fields." Journal of Number Theory 39, no. 3 (1991): 339–50. http://dx.doi.org/10.1016/0022-314x(91)90053-e.

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Ryan, Richard F. "Special forms of the generalized Fermat equation." International Mathematical Forum 17, no. 4 (2022): 201–13. http://dx.doi.org/10.12988/imf.2022.912330.

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Freitas, Nuno, Bartosz Naskręcki, and Michael Stoll. "The generalized Fermat equation with exponents 2, 3,." Compositio Mathematica 156, no. 1 (2019): 77–113. http://dx.doi.org/10.1112/s0010437x19007693.

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We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twis
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DAHMEN, SANDER R. "A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x2 + y2n = z3." International Journal of Number Theory 07, no. 05 (2011): 1303–16. http://dx.doi.org/10.1142/s1793042111004472.

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Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x2 + y2n = z3 in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n ∈ {5,31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n ≤ 107. Finally, we show that there are also no solutions for n ≡ -1 (mod 6).
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Ryan, Richard F. "Solutions to a generalized Fermat equation that has even exponents." International Mathematical Forum 14, no. 6 (2019): 237–46. http://dx.doi.org/10.12988/imf.2019.9835.

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Ivliev, Y. "DIAGNOSTICS OF MATHEMATICAL PROOF OF THE BEAL CONJECTURE IN MEDICAL PSYCHOLOGY (REMAKE OF PREVIOUS AUTHOR’S ARTICLES CONCERNING FERMAT’S LAST THEOREM)." East European Scientific Journal 1, no. 5(69) (2021): 28–33. http://dx.doi.org/10.31618/essa.2782-1994.2021.1.69.48.

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In the given work diagnostics of mathematical proof of the Beal Conjecture (Generalized Fermat’s Last Theorem) obtained in the earlier author’s works was conducted and truthfulness of the suggested proof was established. Realizing the process of the Bill Conjecture solution, the mathematical structure defining hypothetical equality of the Fermat theorem was determined. Such a structure turned to be one of Pythagorean theorem with whole numbers. With help of Euclid’s geometrical theorem and Fermat’s method of infinite descent one can manage to set that Pythagorean equation in whole numbers repr
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Dissertations / Theses on the topic "Generalized Fermat equation"

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Esmonde, Jody. "Parametric solutions to the generalized Fermat equation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0027/MQ50765.pdf.

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Deconinck, Heline. "The generalized Fermat equation over totally real number fields." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/81893/.

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Barroso, de Freitas Nuno Ricardo. "Some Generalized Fermat-type Equations via Q-Curves and Modularity." Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/91288.

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The main purpose of this thesis is to apply the modular approach to Diophantine equations to study some Fermat-type equations of signature (r; r; p) with r >/= 5 a fixed prime and “p” varying. In particular, we will study equations of the form x(r) + y(r) = Cz(p), where C is an integer divisible only by primes “q” is non-identical to 1; 0 (mod “r”) and obtain explicit arithmetic results for “r” = 5, 7, 13. We start with equations of the form x(5) + y(5) = Cz(p). Firstly, we attach two Frey curves E; F defined over Q(square root 5) to putative solutions of the equation. Then by using the
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Books on the topic "Generalized Fermat equation"

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Generalized Fermat Equation. AuthorHouse, 2015.

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Bruin, N. R. Chabauty methods and covering techniques applied to generalized Fermat equations (CWI Tract, 133). CWI Tract, 2002.

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Book chapters on the topic "Generalized Fermat equation"

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Bennett, Michael, Preda Mihăilescu, and Samir Siksek. "The Generalized Fermat Equation." In Open Problems in Mathematics. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32162-2_3.

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Natarajan, Saradha. "Generalized Fermat Equations and the Conjecture of Erdős." In Infosys Science Foundation Series. Springer Nature Singapore, 2025. https://doi.org/10.1007/978-981-96-2599-4_7.

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"Chapter 22. Fermat’s Last Theorem and Generalized Fermat Equations." In Fearless Symmetry. Princeton University Press, 2008. http://dx.doi.org/10.1515/9781400837779.242.

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Balonin Nikolay and Sergeev Mikhail. "Construction of Transformation Basis for Video and Image Masking Procedures." In Frontiers in Artificial Intelligence and Applications. IOS Press, 2014. https://doi.org/10.3233/978-1-61499-405-3-462.

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This article discusses the procedure of image masking using M-matrixes as an orthogonal basis. An overview of some classical matrix calculation algorithms like Silvester, Paley, Scarpis and other, which are common for video bases, is presented. A typical image masking processing chain is commented. Generalized masking algorithms based on new Mersenne, Euler and Fermat matrixes are considered. Examples of Portraits of Hadamard-Mersenne matrixes of 15thand 63rdorders with the levels of elements {1,-b} are presented. A Hadamard-Mersenne matrixes computing algorithm, including equations and formulas applied to calculate their levels, is described. The feasibility of application of such matrixes in antijamming (error-control) processing and image masking tasks is considered.
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Conference papers on the topic "Generalized Fermat equation"

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Simulik, V., I. Krivsky, and I. Lamer. "Generalized clifford - Dirac algebra and Fermi - Bose duality of the dirac equation." In 2012 International Conference on Mathematical Methods in Electromagnetic Theory (MMET). IEEE, 2012. http://dx.doi.org/10.1109/mmet.2012.6331206.

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Journal articles
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