Results Beyond Fermat's Last Theorem

International Journal of Mathematics Trends and Technology (IJMTT)  
© 2022 by IJMTT Journal  
Volume68 Issue8  
Year of Publication : 2022  
Authors : Jagjit Singh Patyal 

10.14445/22315373/IJMTTV68I8P511 
How to Cite?
Jagjit Singh Patyal, "Results Beyond Fermat's Last Theorem," International Journal of Mathematics Trends and Technology, vol. 68, no. 8, pp. 116128, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTTV68I8P511
Abstract
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}(xy)^{n}, where x,y are variable positive integers and x>y, has been analyzed to derive some results relating to the Diophantine equation a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s}, where a,a_{1},a_{2},..., a_{s} 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 a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s}. A result as a theorem 2.1 has been given to find the least positive integral value of s in the equation a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s}. A solution of each of the equations a^{2}=a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n} and a^{3}=a^{3}_{1}+a^{3}_{2}+a^{3}_{3}+a^{3}_{4} has been obtained. It has been proved that the equation a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s} can be expressed as (u+v)^{n}(uv)^{n}=(2a_{2})^{n}+...+(2a_{s})^{n}, where u+v=2a, u+v=2a_{1}. It will also be shown that the Diophantine equation a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s} is a particular case of the equation (x+y)^{n} = (xy)^{n}+2(n1)x^{n1}y+2(n3)x^{n3}y^{3}+...+2α, α={ yn,(n_{n1})xy^{n1}, 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 a^{n}=a^{n}_{1}+a^{n}_{2}+...+a^{n}_{s} has been analyzed to conclude this paper.
Keywords : Diophantine equation, expression, function, number of terms, positive integer.
Reference
[1] Wiles, “Modular Elliptic Curves and Fermat’s Last Theorem,” Annals of Mathematics, vol. 141, no. 3, pp. 443551, 1995.
[2] R. Taylor, A. Wiles, “Ring Theoretic Properties of Certain Hecke Algebras,” Annals of Mathematics, vol. 141, no. 4, pp. 553572, 1995.
[3] B. Roy, “Fermat’s Last Theorem in the Case n=4," Mathematical Gazette, vol.95, pp. 269271, 2011.
[4] K. Rychlik, “On Fermat’s last theorem for n=5, n=10," Mathematics Magazine, vol. 33, no. 5, pp. 279281, 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. 322329, 2004
[9] R. Jennifer, “The Upside Down Pythagorean Theorem,” Mathematical Gazette, vol. 92, pp. 313317, 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. 5154, 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. 261266, 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. 743746 , 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. 127135,1974.
[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. 825835, 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. 440444, 2008.
[22] A.S. Werebrusow, “On the Equation x5+y5= Az5," Math.Samml., vol.25, pp. 466473, 1905.
[23] G. Frey, “Links between Stable Elliptic Curves and Certain Diophantine Equations”, Annales Universitatis Saraviensis Series Mathematicae, vol. 1, pp. 140, 1986.
[24] M. Waldschmidt, “Open Diophantine problems,” Moscow Mathematical Journal, vol. 4, pp. 245305, 2004.
[25] R. D. Carmichael, “On the Impossibility of Certain Diophantine Equations and Systems of Equations,” American Mathematical Monthly, vol. 20, pp. 213221, 1913.
[26] M. Newman, “A Radical Diphantine Equation,” Journal of Number Theory, vol. 13, no. 4, pp. 495498, 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. 269271, 1999.
[29] D. Zagier, “On the Equation w^{4}+x^{4}+y^{4}= z^{4}," Unpublished Note, 1987.