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Journal articles on the topic 'Fermat's equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

Dolan, Stan. "Pell's equation and Fermat." Mathematical Gazette 96, no. 535 (March 2012): 66–70. http://dx.doi.org/10.1017/s0025557200003971.

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Given that it took until the 19th century for the case n = 5 of Fermat's last theorem to be settled, it is not surprising that Fermat's claim of having a proof for all exponents greater than 2 is nowadays treated with considerable scepticism. However, perhaps the most important aspect of his claim has been the impetus it has given to the development of mathematical techniques over the three and a half centuries leading up to the proof by Wiles and Taylor [1, 2]. Not least amongst these techniques has been Fermat's own idea of proof by descente infinie [3,4].

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2

Dem'yanenko, V. A. "ELLIPTIC FUNCTIONS AND FERMAT'S EQUATION." Russian Academy of Sciences. Sbornik Mathematics 77, no. 1 (February 28, 1994): 11–23. http://dx.doi.org/10.1070/sm1994v077n01abeh003426.

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3

Zhong, Cui-Xiang. "On Fermat's equation with prime power exponents." Acta Arithmetica 59, no. 1 (1991): 83–86. http://dx.doi.org/10.4064/aa-59-1-83-86.

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4

Cohn, J. H. E. "The Diophantine equation x2+3 = yn." Glasgow Mathematical Journal 35, no. 2 (May 1993): 203–6. http://dx.doi.org/10.1017/s0017089500009757.

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Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

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5

Kolyvagin, V. A. "Fermat's equation over the tower of cyclotomic fields." Izvestiya: Mathematics 65, no. 3 (June 30, 2001): 503–41. http://dx.doi.org/10.1070/im2001v065n03abeh000337.

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6

Le, Maouha, and Ching Li. "On Fermat's equation in integral 2×2 matrices." Periodica Mathematica Hungarica 31, no. 3 (December 1995): 219–22. http://dx.doi.org/10.1007/bf01882197.

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7

Eynden, Charles Vanden. "Fermat's Last Theorem: 1637—1988." Mathematics Teacher 82, no. 8 (November 1989): 637–40. http://dx.doi.org/10.5951/mt.82.8.0637.

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Around 1637 the French jurist and amateur mathematician Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetic that he had a “truly marvelous” proof that the equation xn + yn = zn has no solution in positive integers if n > 2. Unfortunately the margin was too narrow to contain it. In 1988 the world thought that the Japanese mathematician Yoichi Miyaoka, working at the Max Planck Institute in Bonn, West Germany, might have discovered a proof of this theorem. Such a proof would be of considerable interest because no evidence has been found that Fermat ever wrote one down, and no one has been able to find one in the 350 years since. In fact Miyaoka's announcement turned out to be premature, and a few weeks later articles reported holes in his argument that could not be repaired.

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8

Joseph, James E. "ALGEBRAIC PROOFS FERMAT'S LAST THEOREM, BEAL'S CONJECTURE." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 9 (September 27, 2016): 6576–77. http://dx.doi.org/10.24297/jam.v12i9.130.

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In this paper, the following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; then x; y; z are all even. Also, in this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:

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9

Ribenboim, Paulo. "Fermat's equation for matrices or quaternions over q-adic fields." Acta Arithmetica 113, no. 3 (2004): 241–50. http://dx.doi.org/10.4064/aa113-3-2.

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10

Mestechkin, M. "On periodic continued fractions, Pell equation, and Fermat's challenge numbers." Journal of Computational Methods in Sciences and Engineering 10, no. 1-2 (November 19, 2010): 49–66. http://dx.doi.org/10.3233/jcm-2010-0261.

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11

Seetharaman, P. "A Proof of Fermat's Last Theorem using an Euler's Equation." Asian Research Journal of Mathematics 6, no. 3 (January 10, 2017): 1–24. http://dx.doi.org/10.9734/arjom/2017/36405.

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12

Anglès, Bruno. "Norm Residue Symbol and the First Case of Fermat's Equation." Journal of Number Theory 91, no. 2 (December 2001): 297–311. http://dx.doi.org/10.1006/jnth.2001.2693.

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13

Gandhi, K. Raja Rama, Reuven Tint, and Michael Tint. "Some Anti-Solutions of the Pillai's Conjecture and Proof of Fermat's Last Theorem." Bulletin of Society for Mathematical Services and Standards 8 (December 2013): 26–34. http://dx.doi.org/10.18052/www.scipress.com/bsmass.8.26.

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Let us show several; including the part already well known variants of find uncountable set solutions of equation Axm - Byn = C[1] for natural numbers A,B,C,x,y,m,n of specified values A, B, C, in contrast to the Pillai's conjecture, in which it is assumed that the set of solutions of equation [1] is finite, and proof of Fermat's Last Theorem.

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14

Grytczuk, Aleksander. "On Fermat's equation in the set of integral 2×2 matrices." Periodica Mathematica Hungarica 30, no. 1 (February 1995): 67–72. http://dx.doi.org/10.1007/bf01876927.

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15

Seetharaman, P. "A Proof of Fermat's Last Theorem using an Euler's Equation: A Corollary." Asian Research Journal of Mathematics 9, no. 4 (May 1, 2018): 1–24. http://dx.doi.org/10.9734/arjom/2018/40932.

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16

Shen, Wenda, Jufang Zhang, Shitao Wang, and Shitong Zhu. "Fermat's principle, the general eikonal equation, and space geometry in a static anisotropic medium." Journal of the Optical Society of America A 14, no. 10 (October 1, 1997): 2850. http://dx.doi.org/10.1364/josaa.14.002850.

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17

ADACHI, Norio. "The Diophantine Equation $x^2\pm ly^2=z^l$ Connected with Fermat's Last Theorem." Tokyo Journal of Mathematics 11, no. 1 (June 1988): 85–94. http://dx.doi.org/10.3836/tjm/1270134263.

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18

Ţurcaş, George C. "On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one." Journal of Number Theory 209 (April 2020): 516–30. http://dx.doi.org/10.1016/j.jnt.2019.08.011.

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19

Le, Maohua. "Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents." Colloquium Mathematicum 65, no. 2 (1993): 227–29. http://dx.doi.org/10.4064/cm-65-2-227-229.

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20

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) (June 15, 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 representing Fermat’s Last Theorem cannot exist and then the Fermat theorem is true, that is Fermat’s equality in natural numbers does not exist. Thus mental scheme of “demonstratio mirabile”, which Pierre de Fermat mentioned on the margins of Diophantus’s “Arithmetic”, was reconstructed.

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21

Spetzler, Jesper, and Roel Snieder. "The Fresnel volume and transmitted waves." GEOPHYSICS 69, no. 3 (May 2004): 653–63. http://dx.doi.org/10.1190/1.1759451.

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In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

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22

Dolan, Stan. "Fermat and the difference of two squares." Mathematical Gazette 96, no. 537 (November 2012): 480–91. http://dx.doi.org/10.1017/s0025557200005118.

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In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

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23

Kara, Yasemin, and Ekin Ozman. "Asymptotic generalized Fermat’s last theorem over number fields." International Journal of Number Theory 16, no. 05 (November 28, 2019): 907–24. http://dx.doi.org/10.1142/s1793042120500463.

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Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

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24

Freitas, Nuno, and Samir Siksek. "The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields." Compositio Mathematica 151, no. 8 (March 6, 2015): 1395–415. http://dx.doi.org/10.1112/s0010437x14007957.

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Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation $$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$ are the trivial ones satisfying $abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by $K$, implies the asymptotic Fermat’s Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K=\mathbb{Q}(\sqrt{d})$ for a subset of $d\geqslant 2$ having density ${\textstyle \frac{5}{6}}$ among the squarefree positive integers. We can improve this density to $1$ if we assume a standard ‘Eichler–Shimura’ conjecture.

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25

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 equations.

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26

Ducey, Mark J., and Shawn Fraver. "The conic-paraboloid formulae for coarse woody material volume and taper and their approximation." Canadian Journal of Forest Research 48, no. 8 (August 2018): 966–75. http://dx.doi.org/10.1139/cjfr-2018-0064.

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The conic-paraboloid volume equation is receiving increased use with downed coarse woody material (CWM), but the consequences for taper have not been identified mathematically. Requiring that subdivision of a conic-paraboloid yields two smaller conic-paraboloids leads to an exact taper equation intermediate between those of cones and second-order paraboloids. This exact taper equation does not have an explicit inverse, however. An alternative, naive approach does have an explicit inverse, but subdivision does not yield two conic-paraboloids. The exact conic-paraboloid is closely approximated by Fermat’s paraboloid with exponent 7/5. The exact and naive conic-paraboloids match in volume; differences in taper are ≤2.2% of large-end cross-sectional area and ≤5.9% of large-end diameter, while differences in inverse taper are ≤3.7% of total length. Fermat’s paraboloid is always within 1.2% of total volume; differences in taper are ≤0.8% of large-end cross-sectional area and ≤2.0% of large-end diameter, while differences in inverse taper are ≤1.1% of total length. Such differences are negligible given the variety of CWM shapes and practical measurement challenges. Either the exact conic-paraboloid or the corresponding Fermat’s paraboloid provides appropriate equations for estimating the volume and taper of CWM that is intermediate between conical and ordinary paraboloid frusta.

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27

Herrmann, Emanuel, Attila Pethő, and Horst G. Zimmer. "On fermat’s quadruple equations." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 69, no. 1 (December 1999): 283–91. http://dx.doi.org/10.1007/bf02940880.

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28

DĄBROWSKI, ANDRZEJ. "ON A CLASS OF GENERALIZED FERMAT EQUATIONS." Bulletin of the Australian Mathematical Society 82, no. 3 (June 18, 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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29

Freitas, Nuno, Alain Kraus, and Samir Siksek. "Local criteria for the unit equation and the asymptotic Fermat’s Last Theorem." Proceedings of the National Academy of Sciences 118, no. 12 (March 16, 2021): e2026449118. http://dx.doi.org/10.1073/pnas.2026449118.

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Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat’s Last Theorem to hold over F and also, for the nonexistence of solutions to the unit equation over F. For example, if two totally ramifies and three splits completely in F, then the asymptotic Fermat’s Last Theorem holds over F.

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30

Paparella, Pietro. "Perron numbers that satisfy Fermat’s equation." Notes on Number Theory and Discrete Mathematics 27, no. 3 (September 2021): 119–22. http://dx.doi.org/10.7546/nntdm.2021.27.3.119-122.

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In this note, it is shown that if \ell and m are positive integers such that \ell > m, then there is a Perron number \rho such that \rho^n + (\rho + m)^n = (\rho + \ell)^n. It is also shown that there is an aperiodic integer matrix C such that C^n + (C+ m I_n)^n = (C + \ell I_n)^n.

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31

Панчук, К., K. Panchuk, Е. Любчинов, and E. Lyubchinov. "Cyclographic Interpretation and Computer Solution of One System of Algebraic Equations." Geometry & Graphics 7, no. 3 (December 2, 2019): 3–14. http://dx.doi.org/10.12737/article_5dce5e528e4301.77886978.

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The subject of this study is an algebraic equation of one form and a system of such equations. The peculiarity of the subject of research is that both the equation and the system of equations admit a cyclographic interpretation in the operational Euclidean space, the dimension of which is one more than the dimension of the subspace of geometric images described by the original equations or system of equations. The examples illustrate the advantages of cyclographic interpretation as the basis of the proposed solutions, namely: it allows you to get analytical, i.e. exact solutions of the complete system of equations of the considered type, regardless of the dimension of the subspace of geometric objects described by the equations of the system; in the geometric version of the solution of the system (the Apollonius and Fermat problems), no application of any transformations (inversions, circular transforms, etc.) is required, unlike many existing methods and approaches; constructive and analytical solutions of the system of equations, mutually complementary, are implemented by available means of graphic CAD and computer algebra. The efficiency of cyclographic interpretation is shown in obtaining an analytical solution to the Fermat problem using a computer algebra system. The solution comes down to determining in the operational space the points of intersection of the straight line and the 3-α-rotation cone with the semi-angle α = 45° at its vertex. The cyclographic images of two intersection points in the operational space are the two desired spheres in the subspace of given spheres. A generalization of the proposed algorithm for the analytical solution of the Fermat problem for n given (n – 2)-spheres in (n – 1)-dimensional subspace. It is shown that in this case the analytical solution of the Fermat problem is reduced to determining the intersection points of the straight line and the (n – 1)-α-cone of rotation in the operational n-dimensional Euclidean space.

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32

Bennett, Michael A., Imin Chen, Sander R. Dahmen, and Soroosh Yazdani. "Generalized Fermat equations: A miscellany." International Journal of Number Theory 11, no. 01 (November 24, 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 previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

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33

Ghanouchi, Jamel. "A Proof of Beal’s Conjecture." Bulletin of Mathematical Sciences and Applications 5 (August 2013): 30–34. http://dx.doi.org/10.18052/www.scipress.com/bmsa.5.30.

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More than one century after its formulation by the Belgian mathematician Eugene Catalan, Preda Mihailescu has solved the open problem. But, is it all ? Mihailescu's solution utilizes computation on machines, we propose here not really a proof of Catalan theorem as it is entended classically, but a resolution of an equation like the resolution of the polynomial equations of third and fourth degrees. This solution is totally algebraic and does not utilize, of course, computers or any kind of calculation. We generalize our approach to Beal equation and discuss the solutions. (Keywords: Diophantine equations, Catalan, Fermat-Catalan, Conjectures, Proofs, Algebraic resolution).

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34

DAHMEN, SANDER R. "A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x2 + y2n = z3." International Journal of Number Theory 07, no. 05 (August 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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35

Dang, Guoqiang, and Jinhua Cai. "Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation." Journal of Mathematics 2020 (July 7, 2020): 1–8. http://dx.doi.org/10.1155/2020/4871812.

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In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation f″z2+△ckfz2=1 and the system of equations f1″z2+△ckf2z2=1 and f2″z2+△ckf1z2=1 have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

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36

Korotkov, Aleksandr Sergeevich, Anton Victorovich Gyliaev, and Vladimir Aleksandrovich Korotkov. "Method of Equivalent Equations to Prove Fermat's Theorem." Automation and Control in Technical Systems, no. 1 (April 5, 2015): 139. http://dx.doi.org/10.12731/2306-1561-2015-1-16.

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37

Siksek, Samir. "Diophantine equations after Fermat’s last theorem." Journal de Théorie des Nombres de Bordeaux 21, no. 2 (2009): 423–34. http://dx.doi.org/10.5802/jtnb.679.

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38

Tan Si, Do. "Fermat’s Last Theorem proven in one page." Applied Physics Research 10, no. 3 (May 31, 2018): 18. http://dx.doi.org/10.5539/apr.v10n3p20.

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We attempt to prove the Fermat’s Last Theorem by a simple method consisted in transforming the relation b^m=(a+n)^m-a^m into an equation in n by introduction of a parameter ɷ depending in a,n such that b=omega^(m(m-1)) then equalizing in b^m these two relations. Afterward, exploiting the condition that this equation must have only one root so that the coefficients of powers of n^i must have alternating signs, we arrive to conclude that the equation in n has roots only for m=1,2 and no root for m>2 thus prove the theorem.

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39

Korhonen, Risto, and Yueyang Zhang. "Existence of Meromorphic Solutions of First-Order Difference Equations." Constructive Approximation 51, no. 3 (December 17, 2019): 465–504. http://dx.doi.org/10.1007/s00365-019-09491-0.

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AbstractIt is shown that if $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$f(z+1)n=R(z,f),where R(z, f) is rational in f with meromorphic coefficients and $$\deg _f(R(z,f))=n$$degf(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if $$f(z+1)^n=R(z,f)$$f(z+1)n=R(z,f), where the assumption $$\deg _f(R(z,f))=n$$degf(R(z,f))=n has been discarded, has rational coefficients and a transcendental meromorphic solution f of hyper-order $$<1$$<1, then either f satisfies a difference linear or Riccati equation with rational coefficients, or the equation above can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions that are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

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40

Gundersen, Gary G., Katsuya Ishizaki, and Naofumi Kimura. "Restrictions on meromorphic solutions of Fermat type equations." Proceedings of the Edinburgh Mathematical Society 63, no. 3 (May 8, 2020): 654–65. http://dx.doi.org/10.1017/s001309152000005x.

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AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

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CHEN, MIN FENG, and ZONG SHENG GAO. "ENTIRE SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATION AND FERMAT TYPE q-DIFFERENCE DIFFERENTIAL EQUATIONS." Communications of the Korean Mathematical Society 30, no. 4 (October 31, 2015): 447–56. http://dx.doi.org/10.4134/ckms.2015.30.4.447.

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Schuster, Gerard T., and Min Zhou. "A theoretical overview of model-based and correlation-based redatuming methods." GEOPHYSICS 71, no. 4 (July 2006): SI103—SI110. http://dx.doi.org/10.1190/1.2208967.

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We review the equations for correlation-based redatuming methods. A correlation-based redatuming method uses natural-phase information in the data to time shift the weighted traces so they appear to be generated by sources (or recorded by geophones) shifted to a new location. This compares to model-based redatuming, which effectively time shifts the traces using traveltimes computed from a prior velocity model. For wavefield redatuming, the daylight imaging, interferometric imaging, reverse-time acoustics (RTA), and virtual-source methods all require weighted correlation of the traces with one another, followed by summation over all sources (and sometimes receivers). These methods differ from one another by their choice of weights. The least-squares interferometry and virtual-source imaging methods are potentially the most powerful because they account for the limited source and receiver aperture of the recording geometry. Interferometry, on the other hand, has the flexibility to select imaging conditions that target almost any type of event. Stationary-phase principles lead to a Fermat-based redatuming method known as redatuming by a seminatural Green’s function. No crosscorrelation is needed, so it is less expensive than the other methods. Finally, Fermat’s principle can be used to redatum traveltimes.

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Fine, Benjamin. "Cyclotomic equations and square properties in rings." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 89–95. http://dx.doi.org/10.1155/s016117128600011x.

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IfRis a ring, the structure of the projective special linear groupPSL2(R)is used to investigate the existence of sum of square properties holding inR. Rings which satisfy Fermat's two-square theorem are called sum of squares rings and have been studied previously. The present study considers a related property called square property one. It is shown that this holds in an infinite class of rings which includes the integers, polynomial rings over many fields andZpnwherePis a prime such that−3is not a squaremodp. Finally, it is shown that the class of sum of squares rings and the class satisfying square property one are non-coincidental.

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Pila, Jonathan. "On a modular Fermat equation." Commentarii Mathematici Helvetici 92, no. 1 (2017): 85–103. http://dx.doi.org/10.4171/cmh/407.

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45

Lenstra, H. W. "On the inverse Fermat equation." Discrete Mathematics 106-107 (September 1992): 329–31. http://dx.doi.org/10.1016/0012-365x(92)90561-s.

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46

Levesque, C. "On a few Diophantine equations, in particular, Fermat's last theorem." International Journal of Mathematics and Mathematical Sciences 2003, no. 71 (2003): 4473–500. http://dx.doi.org/10.1155/s0161171203210668.

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This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solvedà laWiles. We will exhibit many families of Thue equations, for which Baker's linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, theABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

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Karmakar, Sudhangshu B. "An elementary proof of Fermat’s last theorem for all even exponents." Journal of Mathematical Cryptology 14, no. 1 (July 3, 2020): 139–42. http://dx.doi.org/10.1515/jmc-2016-0018.

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AbstractAn elementary proof that the equation x2n + y2n = z2n can not have any non-zero positive integer solutions when n is an integer ≥ 2 is presented. To prove that the equation has no integer solutions it is first hypothesized that the equation has integer solutions. The absence of any integer solutions of the equation is justified by contradicting the hypothesis.

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Teia, Luis. "Fermat's Theorem -- a Geometrical View." Journal of Mathematics Research 9, no. 1 (January 23, 2017): 136. http://dx.doi.org/10.5539/jmr.v9n1p136.

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Fermat's Last Theorem questions not only what is a triple, but more importantly, what is an integer in the context of equations of the type $x^n+y^n=z^n$. This paper explores these questions in one, two and three dimensions. It was found that two conditions are required for an integer element to exist in the context of the Pythagoras' theorem in 1D, 2D and 3D. An integer must satisfy the Pythagoras' theorem of the respective dimension -- condition 1. And, it must be completely successfully split into multiple unit scalars -- condition 2. In 1D, the fundamental unit scalar is the line length 1. All integers in 1D satisfy $x+y=z$, and can be decomposed into multiples of the unit line, hence integers exist and can form 1D triples $(x,y,z)$. In 2D, the fundamental unit scalar is the square side 1. Only some groups of integers (called triples) satisfy $x^2+y^2=z^2$, and can be decomposed into multiples of the unit square, forming 2D triples. In 3D, the fundamental unit scalar is the octahedron side 1. The geometry of the 3D Pythagoras' theorem dictates that $x^3+y^3=z^3$ is governed by octahedrons, validating condition 1. However, octahedrons with side length integer cannot be completely divided into unit octahedrons (as tetrahedrons appear), invalidating condition 2. Hence, if integers do not exist in the context of the 3D Pythagoras' theorem, then neither do triples. This confirms Fermat's Last Theorem for three dimensions ($n=3$). The geometrical interdependency between integers in 1D and 2D suggests that all integers of higher dimensions are built, and hence are dependent, on the integers of lower dimensions. This interdependency coupled with the absence of integers in 3D suggests that there are no integers above $n>2$, and therefore there are also no triples that satisfy $x^n+y^n=z^n$ for $n>2$.

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Ghanouchi, Jamel. "An Elementary Proof of Fermat-Wiles Theorem and Generalization to Beal Conjecture." Bulletin of Society for Mathematical Services and Standards 12 (December 2014): 4–9. http://dx.doi.org/10.18052/www.scipress.com/bsmass.12.4.

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Ghanouchi, Jamel. "An Elementary Proof of Fermat-Wiles and Catalan-Mihailescu Theorems and Generalization to Beal Conjecture." Bulletin of Mathematical Sciences and Applications 8 (May 2014): 1–5. http://dx.doi.org/10.18052/www.scipress.com/bmsa.8.1.

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