Glasnik Matematicki, Vol. 60, No. 1 (2025), 21-38. \( \)

A NOTE ON SOME POLYNOMIAL-FACTORIAL DIOPHANTINE EQUATIONS

Saša Novaković

Hochschule Fresenius University of applied Sciences, 40476 Düsseldorf, Germany
e-mail:sasa.novakovic@hs-fresenius.de

Abstract.   In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation \(n!=x^2-1\). It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed that a weak form of Szpiro's-conjecture implies that Brocard's equation has finitely many integer solutions. More generally, assuming the ABC-conjecture, Luca showed that equations of the form \(n!=P(x)\) where \(P(x)\in\mathbb{Z}[x]\) of degree \(d\geq 2\) have only finitely many integer solutions with \(n\gt 0\). And if \(P(x)\) is irreducible, Berend and Harmse proved unconditionally that \(P(x)=n!\) has only finitely many integer solutions. In this note we study Diophantine equations of the form \(g(x_1,\ldots,x_r)=P(x)\) where \(P(x)\in\mathbb{Z}[x]\) of degree \(d\geq 2\) and \(g(x_1,\ldots,x_r)\in \mathbb{Z}[x_1,\ldots,x_r]\) where for the \(x_i\) one may also plug in \(A^{n}\) or the Bhargava factorial \(n!_S\). We want to understand when there are finitely many or infinitely many integer solutions. Moreover, we study Diophantine equations of the form \(g(x_1,\ldots,x_r)=f(x,y)\) where \(f(x,y)\in\mathbb{Z}[x,y]\) is a homogeneous polynomial of degree \(\geq2\).

2020 Mathematics Subject Classification.   11D61, 11D72, 11D59, 14G05

Key words and phrases.   Diophantine equations, factorials, polynomials

https://doi.org/10.3336/gm.60.1.02

References:

1. D. Berend and J. E. Harmse, On polynomial-factorial Diophantine equations, Trans. Amer. Math. Soc. 358 (2006), 1741–1779.
   MathSciNet    CrossRef

2. D. Berend and C. F. Osgood, On the equation \(P(x)=n!\) and a question of Erdős, J. Number Theory 42 (1992), 189–193.
   MathSciNet    CrossRef

3. B. C. Berndt and W. F. Galeway, On the Brocard-Ramanujan Diophantine equation \(n!+1=m^2\), Ramanujan J. 4 (2000), 41–42.
   MathSciNet    CrossRef

4. M. Bhargava, P-orderings and polynimial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101–127.
   MathSciNet    CrossRef

5. H. Brocard, Question 1532, Nouv. Ann. Math. 4 (1885), 391.

6. Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential and Diophantine equations II. The Lebesgue–Nagel equation, Compos. Math. 142 (2006), 31–62.
   MathSciNet    CrossRef

7. H. M. Bui, K. Pratt and A. Zaharescu, Power savings for counting solutions to polynomial-factorial equations, Adv. Math. 422 (2023), Paper No. 109021, 32 pp.
   MathSciNet    CrossRef

8. A. Da̧browski, On the Diophantine equation \(x!+A=y^2\), Nieuw Arch. Wisk. 14 (1996), 321–324.
   MathSciNet

9. A. Da̧browski, On the Brocard–Ramanujan problem and generalizations, Colloq. Math. 126 (2012), 105–110.
   MathSciNet    CrossRef

10. A. Da̧browski and M. Ulas, Variations on the Brocard–Ramanujan equation, J. Number Theory 133 (2013), 1168–1185.
    MathSciNet    CrossRef

11. A. Epstein and J. Glickman, 2020, .

12. P. Erdős and R. Obláth, Über Diophantische Gleichungen der Form \(n!=x^p \pm y^p\) und \(n!\pm m!= x^p\), Acta Szeged. 8 (1937), 241–255.

13. O. Kihel and F. Luca, Variants of the Brocard–Ramanujan equation, J. Théor. Nombres Bordeaux 20 (2008), 353–363.
    MathSciNet    CrossRef

14. S. Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. 23 (1990), 37–75.
    MathSciNet    CrossRef

15. F. Luca, The Diophantine equation \(P(x)=n!\) and a result of M. Overholt, Glas. Mat. Ser. III 37(57) (2002), 269–273.
    MathSciNet

16. R. D. Matson, Brocard's problem 4th solution search utilizing quadratic residues, Unsolved Problems in Number Theory, Logic and Cryptography (2017), available at http://unsolvedproblems.org/S99.pdf

17. M. Overholt, The Diophantine equation \(n!+1=m^2\), Bull. London. Math. Soc. 25 (1993), 104.
    MathSciNet    CrossRef

18. R. M. Pollack and H. N. Shapiro, The next to last case of a factorial Diophantine equation, Comm. Pure Appl. Math. 26 (1973), 313–325.
    MathSciNet    CrossRef

19. S. Ramanujan, Question 469, J. Indian Math. Soc. 5 (1913), 59.

20. T. N. Shorey, Diophantine approximations, Diophantine equations, transcendence and applications, Indian J. Pure Appl. Math. 37 (2006), 9–39.
    MathSciNet

21. T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Math. 87, Cambridge University Press, Cambridge, 1986.
    MathSciNet    CrossRef

22. S. Siksek, Diophantine equations after Fermat's last theorem, J. Théor. Nombres Bordeaux 21 (2009), 425–436.
    MathSciNet    Link

23. W. Takeda, On the finiteness of solutions for polynomial-factorial Diophantine equations, Forum Math. 33 (2021), 361–374.
    MathSciNet    CrossRef

24. M. Ulas, Some observations on the Diophantine equation \(y^2=x!+A\) and related results, Bull. Aust. Math. Soc. 86 (2012), 377–388.
    MathSciNet    CrossRef

25. M. Ulas, Some experiments with Ramanujan–Nagell type Diophantine equations, Glas. Mat. Ser. III 49(69) (2014), 287–302.
    MathSciNet    CrossRef

26. T. Yamada, A generalization of the Ramanujan–Nagell equation, Glasg. Math. J. 61 (2019), 535–544.
    MathSciNet    CrossRef