Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:02:58.076Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME OBSERVATIONS ON THE DIOPHANTINE EQUATION y2=x!+A AND RELATED RESULTS

Part of: Diophantine equations

Published online by Cambridge University Press:  02 August 2012

MACIEJ ULAS
MACIEJ ULAS*
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6, 30-348 Kraków, Poland (email: maciej.ulas@uj.edu.pl)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the Brocard–Ramanujan type Diophantine equation y2=x!+A and ask about values of A∈ℤ for which there are at least three solutions in the positive integers. In particular, we prove that the set 𝒜 consisting of integers with this property is infinite. In fact we construct a two-parameter family of integers contained in 𝒜. We also give some computational results related to this equation.

Keywords

factorial functionBrocard–Ramanujan type equationDiophantine equation

MSC classification

Secondary: 11D85: Representation problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 377 - 388
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Berend, D. and Harmse, J. E., ‘On polynomial-factorial diophantine equations’, Trans. Amer. Math. Soc. 358 (2005), 1741–1779.CrossRefGoogle Scholar
[2]Berndt, B. C. and Galway, W. F., ‘On the Brocard–Ramanujan Diophantine equation n!+1=m 2’, Ramanujan J. 4 (2000), 41–42.CrossRefGoogle Scholar
[3]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24(3–4) (1997), 235–265.CrossRefGoogle Scholar
[4]Brocard, H., ‘Question 166’, Nouv. Corresp. Math. 2 (1876), 287.Google Scholar
[5]Brocard, H., ‘Question 1532’, Nouv. Ann. Math. 4 (1885), 291.Google Scholar
[6]Da̧browski, A., ‘On the Diophantine equation x!+A=y 2’, Nieuw Arch. Wiskd. (4) 14(3) (1996), 321–324.Google Scholar
[7]Da̧browski, A., ‘On the Brocard–Ramanujan problem and generalizations’, Colloq. Math. 126 (2012), 105–110.CrossRefGoogle Scholar
[8]Guy, R. K., Unsolved Problems in Number Theory (Springer, New York, 1994).CrossRefGoogle Scholar
[9]Kihel, O. and Luca, F., ‘Variants of the Brocard-Ramanujan equation’, J. Théor. des Nombres Bordeaux 20 (2008), 353–363.CrossRefGoogle Scholar
[10]Kihel, O., Luca, F. and Togbé, A., ‘Variant of the Diophantine equation n!+1=y 2’, Port. Math. 67 (2010), 1–11.CrossRefGoogle Scholar
[11]Luca, F. and Togbe, A., ‘On numbers of the form ±x 2±y!’, in: Proc. Conference Diophantine Equations, 2005 (Narosa Publishing House, New Delhi, 2007), pp. 135–150.Google Scholar
[12]Overholt, M., ‘The Diophantine equation n!+1=m 2’, Bull. Lond. Math. Soc. 25 (1993), 104.CrossRefGoogle Scholar
[13]Ramanujan, S., ‘Question 469’, J. Indian Math. Soc. 5 (1913), 59.Google Scholar
[14]Togbé, A., ‘A note on the Diophantine equation x!+A=y 2’, Int. J. Pure Appl. Math. 26(4) (2006), 565–571.Google Scholar
[15]Togbé, A., ‘A note on the Diophantine equation x!+A=y 2, II’, Int. J. Appl. Math. Stat. 26(6) (2006), 25–32.Google Scholar