Glasnik Matematicki, Vol. 48, No. 1 (2013), 31-48.


Florian Luca

Fundación Marcos Moshinsky, Universidad Nacional Autonoma de México, Circuito Exterior, C.U., Apdo. Postal 70-543, Mexico D.F. 04510, Mexico

Abstract.   In this paper, we show under the abc conjecture that the Diophantine equation f(x)=u!+v! has only finitely many integer solutions (x,u,v) whenever f(X) Q[X] is a polynomial of degree at least three.

2010 Mathematics Subject Classification.   11D85.

Key words and phrases.   Factorials, polynomials, applications of the abc conjecture.

Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.48.1.03


  1. C. Ballot and F. Luca, Prime factors of af(n)-1 with an irreducible polynomial f(x), New York J. Math. 12 (2006), 39-45.
    MathSciNet     CrossRef

  2. D. Berend and J. Harmse, On polynomial factorial diophantine equations, Trans. Amer. Math. Soc. 358 (2005), 1741-1779.
    MathSciNet     CrossRef

  3. P. Erdös and R. Obláth, Uber diophantishe Gleichungen der Form n! = xp ± yp und n! ± m! = xp, Acta Litt. Sci. Szeged 8 (1937), 241-255.

  4. M. Gawron, On the equation P(z)=n!+m!, preprint.

  5. A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107.
    MathSciNet     CrossRef

  6. H. Iwaniec and J. Pintz, Primes in short intervals, Monatsh. Math. 98 (1984), 115-143.
    MathSciNet     CrossRef

  7. F. Luca, The Diophantine equation P (x) = n! and a result of M. Overholt, Glas. Mat. Ser. III 37 (2002), 269-273.

Glasnik Matematicki Home Page