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11-XX Number theory
11Dxx Diophantine equations

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On $x^{n} + y^{n} = n! z^{n}$

Susil Kumar Jena (2018)

Communications in Mathematics

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 $x^{n} + y^{n} = n! z^{n}$ has no integer solutions with $n \in ℕ_{+}$ and $n > 2$ . But, contrary to this expectation, we show that for $n = 3$ , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd (x, y, z) = 1$ .

On x³ + y³ + z³ = 3μxyz and Jacobi polynomials

Kaori Ota (1994)

Acta Arithmetica

On zeros of diagonal forms over p-adic fields

Yismaw Alemu (1987)

Acta Arithmetica

On zeros of forms over local fields

Yismaw Alemu (1985)

Acta Arithmetica

One Erdös style inequality

Tomáš J. Kepka, Petr C. Němec (2019)

Commentationes Mathematicae Universitatis Carolinae

One unusual inequality is examined.

Original title unknown

А.Б. Воронецкий (1986)

Zapiski naucnych seminarov Leningradskogo

Orthant spanning simplexes with minimal volume.

Elia, Michele (2003)

International Journal of Mathematics and Mathematical Sciences

Osservazioni sul problema di Fermat.

Alpinolo Natucci (1951)

Bollettino dell'Unione Matematica Italiana

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