Volume 68 | Issue 8 | Year 2022 | Article Id. IJMTT-V68I8P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I8P511

Received | Revised | Accepted | Published |
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05 Jul 2022 | 07 Aug 2022 | 19 Aug 2022 | 27 Aug 2022 |

In this paper, some results relating to Fermat's last theorem and beyond this theorem, have been presented. The expression of the form (x+y)n-(x-y)n, where x,y are variable positive integers and x>y, has been analyzed to derive some results relating to the Diophantine equation an=an1+an2+...+ans, where a,a1,a2,..., as are positive integers. An attempt has been made to give a simple proof of Fermat's last theorem and further this theorem has been extended to the case of s=3 relative to the equation an=an1+an2+...+ans. A result as a theorem 2.1 has been given to find the least positive integral value of s in the equation an=an1+an2+...+ans. A solution of each of the equations a2=a21+a22+...+a2n and a3=a31+a32+a33+a34 has been obtained. It has been proved that the equation an=an1+an2+...+ans can be expressed as (u+v)n-(u-v)n=(2a2)n+...+(2as)n, where u+v=2a, u+v=2a1. It will also be shown that the Diophantine equation an=an1+an2+...+ans is a particular case of the equation (x+y)n = (x-y)n+2(n1)xn-1y+2(n3)xn-3y3+...+2α, α={ yn,(nn-1)xyn-1, if n is odd if n is even as it is obtained by putting some positive integeral values u,v(u>v) of x,y respectively. Finally equation an=an1+an2+...+ans has been analyzed to conclude this paper.

[1] Wiles, “Modular Elliptic Curves and Fermat’s Last Theorem,” Annals of Mathematics, vol. 141, no. 3, pp. 443-551, 1995.

[2] R. Taylor, A. Wiles, “Ring Theoretic Properties of Certain Hecke Algebras,” Annals of Mathematics, vol. 141, no. 4, pp. 553-572, 1995.

[3] B. Roy, “Fermat’s Last Theorem in the Case n=4," Mathematical Gazette, vol.95, pp. 269-271, 2011.

[4] K. Rychlik, “On Fermat’s last theorem for n=5, n=10," Mathematics Magazine, vol. 33, no. 5, pp. 279-281, 1960.

[6] L.M. Adleman, D.R. Heath Brown, “The First Case of the Fermat Last Theorem,” Inventiones Mathematica, Berlin, Springer, vol. 79, no. 2, 1985.

[7] H. M. Edwards, “Fermat Last Theorem: A Genetic Introduction to Number Theory, Graduate Texts in Mathematics,” Springer Verlag, vol. 50, pp. 79, 1977.

[8] C. D. Bennett, A.M.W. Glass, Szekely and J. Gaber, “Fermat’s Last Theorem for Rational Exponents,” Americal Mathematical Monthly, vol. 111, no. 4, pp. 322-329, 2004

[9] R. Jennifer, “The Upside Down Pythagorean Theorem,” Mathematical Gazette, vol. 92, pp. 313-317, 2008.

[10] A. Van der Poortan , “Notes on Fermat’s Last Theorem”, 1996.

[11] P. Ribenboim, “13 Lectures on Fermat’s Last Theorem,” Springer Verlag, pp. 51-54, 1979.

[12] S. Singh, “Fermat’s Enigma,” Anchor Books, 1998.

[13] M. Charles, “The Fermat’s Diary,” American Mathematical Society, 2000.

[14] G.Cornell , J.H. Silverman and G. Stevens , “Modular Forms and Fermat’s Last Theorem”, Springer, pp. 482, 1997.

[15] K. Buzzard , “Review of Modular Forms and Fermat’s Last Theorem by G. Cornell, J.H. Silverman and G. Stevens,” Bulletin of the American Mathematical Society, vol. 36, no. 2, pp. 261-266, 1999.

[16] G. Faltings, “The proof of Fermat’s last theorem by R Taylor and A. Wiles,” Notices of the American Mathematical Society, vol. 42, no.7, pp. 743-746 , 1995.

[17] A. Aczel, “Fermat’s Last Theorem: Unlocking the Secret of Ancient Mathematical Problem,” 1997.

[18] V.A. Demjanenko, “L. Eulers Conjecture,” Acta Arith, vol. 25, pp. 127-135,1974.

[19] L.J. Lander and T. R. Parkin, “Counter Examples to the Euler’s Conjecture on the Sums of Like Powers,” Bull. Amer. Math . Soc., vol. 72, pp. 1079, 1966.

[20] N. D. Elkies, “On A4+B4+C4 = D4," Math.. Comput.. vol. 51, pp. 825-835, 1988.

[21] G. H. Hardy, E. M. Wright revised by D. R. Heath Brown, J. H. Silverman, “An Introduction to the Theory of Numbers,” 6th ed., Oxford University Press, pp. 260, pp. 440-444, 2008.

[22] A.S. Werebrusow, “On the Equation x5+y5= Az5," Math.Samml., vol.25, pp. 466-473, 1905.

[23] G. Frey, “Links between Stable Elliptic Curves and Certain Diophantine Equations”, Annales Universitatis Saraviensis Series Mathematicae, vol. 1, pp. 1-40, 1986.

[24] M. Waldschmidt, “Open Diophantine problems,” Moscow Mathematical Journal, vol. 4, pp. 245-305, 2004.

[25] R. D. Carmichael, “On the Impossibility of Certain Diophantine Equations and Systems of Equations,” American Mathematical Monthly, vol. 20, pp. 213-221, 1913.

[26] M. Newman, “A Radical Diphantine Equation,” Journal of Number Theory, vol. 13, no. 4, pp. 495-498, 1981.

[27] L.E. Dickson, “History of the Theory of Numbers, Diophantine analysis,” G,E. Stechert & Co., vol. 2, 1934.

[28] V. Roger, “ Integers Solution of a^{-2}+b^{-2}=d^{-2}," Mathematical Gazette, vol. 83, no. 497, pp. 269-271, 1999.

[29] D. Zagier, “On the Equation w^{4}+x^{4}+y^{4}= z^{4}," Unpublished Note, 1987.

Jagjit Singh Patyal, "Results Beyond Fermat's Last Theorem," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 68, no. 8, pp. 116-128, 2022. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V68I8P511