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

Abstract

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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