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On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2

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Abstract

In 1876, H. Brocard posed the problem of finding all integral solutions to n! + 1 = m2. In 1913, unaware of Brocard's query, S. Ramanujan gave the problem in the form, “The number 1 + n! is a perfect square for the values 4, 5, 7 of n. Find other values.” We report on calculations up to n = 109 and briefly discuss a related problem.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois, 61801

    Bruce C. Berndt

  2. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois, 61801

    William F. Galway

Authors
  1. Bruce C. Berndt
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  2. William F. Galway
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Berndt, B.C., Galway, W.F. On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2. The Ramanujan Journal 4, 41–42 (2000). https://doi.org/10.1023/A:1009873805276

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  • DOI: https://doi.org/10.1023/A:1009873805276

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