Published July 30, 2024 | Version CC-BY-NC-ND 4.0
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Solution of Brocard's Problem

Creators

  • 1. Burlington, Ontario, Canada.

Description

Abstract: Brocard's problem is the solution of the equation, 𝒏!+𝟏= π’ŽπŸ, where m and n are natural numbers. So far only 3solutions have been found, namely (n,m) = (4,5), (5,11), and (7,71). The purpose of this paper is to show that there are no other solutions. Firstly, it will be shown that if (n,m) is to be a solution to Brocard's problem, then n! = 4AB, where A is even, B is odd, and |A – B| = 1. If n is even (n = 2x) and > 4, it will be shown that necessarily 𝑨=(πŸπ’™)β€ΌπŸ’π’š and 𝑩=π’š(πŸπ’™−𝟏)β€Ό, for some odd y > 1. Next, it will be shown that x < 2y, and this leads to an inequality in x [namely, (𝒙(πŸπ’™−𝟏)β€Ό ± 𝟏)𝟐−𝟏−(πŸπ’™)!<𝟎],for which there is no solution when x ≥ 3. If n is odd, there is a similar procedure.

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Additional details

Identifiers

DOI
10.54105/ijam.B1174.04010424
EISSN
2582-8932

Dates

Accepted
2024-04-15
Manuscript received on 09 March 2024 | Revised Manuscript received on 16 March 2024 | Manuscript Accepted on 15 April 2024 | Manuscript published on 30 April 2024.

References

  • Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287
  • Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391
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  • Wikipedia: https://en.wikipedia.org/wiki/Brocard%27s_problem
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