On xn + yn = n!zn

  • 1 Former Professor of Electronics and Telecommunication Engineering Plot No-170, Sector-5, Niladri Vihar, Chandrasekharpur, Bhubaneswar-751021,, Odisha, India


In p. 219 of R.K. Guy's Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation xn + yn = n!zn has no integer solutions with n ∈ N+ and n > 2. But, contrary to this expectation, we show that for n = 3, this equation has in finitely many primitive integer solutions, i.e. the solutions satisfying the condition gcd(x, y, z) = 1.

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  • [1] N. D. Elkies: Wiles minus epsilon implies Fermat. Elliptic Curves, Modular Forms & Fermat's Last Theorem (1995) 38{40. Ser. Number Theory, I, Internat. Press, Cambridge MA.

  • [2] P. Erdős, R. Obláth: Über diophantische Gleichungen der form n! = xp ±yp and n! ± m! = xp. Acta Litt. Sci. Szeged 8 (1937) 241-255.

  • [3] R. K. Guy: Unsolved Problems in Number Theory. Springer Science+Business Media, Inc., New York (2004). Third Edition.

  • [4] K. Ribet: On modular representations of Gal(Q n Q) arising from modular forms. Invent. Math. 100 (1990) 431-476.


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