On xn + yn = n!zn

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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.

[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.

Journal Information

CiteScore 2017: 0.33

SCImago Journal Rank (SJR) 2017: 0.128
Source Normalized Impact per Paper (SNIP) 2017: 0.476

Mathematical Citation Quotient (MCQ) 2017: 0.43

Target Group

researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory


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