Elementary proof of W. Schulte’s conjecture

Djamel Himane
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 251–255
DOI: 10.7546/nntdm.2025.31.2.251-255
Full paper (PDF, 187 Kb)

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Authors and affiliations

Djamel Himane
LA3C Laboratory, Faculty of Mathematics, USTHB
P. O. Box 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

Abstract

In the On-Line Encyclopedia of Integer Sequences, we find the sequence 1, 1, 3, 18, 180, 2700, 56700, 1587600, 57153600, \dots, which is given by the formula A_{n} = n!(n-1)!/2^{n-1}. On the same page, Werner Schulte conjectured that for all n > 1, n divides 2A_{n-1} + 4 if and only if n is prime. In this paper, we employ elementary methods to provide a simple proof of this conjecture.

Keywords

  • Legendre’s formula
  • Wilson’s theorem
  • Fermat’s Little Theorem

2020 Mathematics Subject Classification

  • 11A41
  • 11A51

References

  1. Cassels, J. W. S., & Fröhlich, A. (1967). Algebraic Number Theory. Academic Press, p. 9.
  2. Legendre, A. M. (1830). Théorie des Nombres. Paris, p. 10.
  3. Ribenboim, P. (1996). The New Book of Prime Number Records. Springer Verlag.
  4. Sloane, N. J. A. A006472 a(n) = n!\times(n-1)!/2^{(n-1)}. (Formerly M3052). The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org/A006472.
  5. Sloane, N. J. A., & Plouffe, S. (1995). The Encyclopedia of Integer Sequences. Academic Press, p. 319.

Manuscript history

  • Received: 14 November 2024
  • Accepted: 8 May 2025
  • Online First: 9 May 2025

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Himane, D. (2025). Elementary proof of W. Schulte’s conjecture. Notes on Number Theory and Discrete Mathematics, 31(2), 251-255, DOI: 10.7546/nntdm.2025.31.2.251-255.

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