Naciri 2025 brocard ramanujan equation free integers prime

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A. M. Naciri, On the Brocard-Ramanujan equation with $7$-free integers and prime powers. Integers 25 (2025), #A71. No license line is printed in the file (p. 1 carries "#A71 INTEGERS 25 (2025)" and "DOI: 10.5281/zenodo.16881781"); the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02), the Creative Commons Attribution 4.0 license.

Theorem 1 proves two finiteness statements for the Brocard-Ramanujan equation n! + 1 = x^2: (i) for each k >= 2 there are only finitely many solutions with x+1 or x-1 a k-free number, and for k = 7 the only possible solutions are the three known pairs (4,5), (5,11), (7,71); (ii) for each l >= 2 there are only finitely many solutions with x±1 having fewer than l prime divisors, and when x±1 is a prime power the only possible solution is (4,5). The proofs are elementary analytic, combining the Chebyshev-type upper bound pi(n) <= (3/2) n/ln n (p. 2 prints it with the lower bound n/ln n <= pi(n) "for all n >= 2", which fails for small n, e.g. n = 2; only the upper bound is used), Stirling's estimate (n/e)^n <= n! <= n^n, and Legendre's formula for nu_p(n!) to show that the factorization of n! = (x-1)(x+1) forces too many prime powers to fit inside a k-free or few-prime-factor value. Section 4 gives a generalization of Theorem 1 (Theorem 2) and a remark on what a full resolution of the Brocard-Ramanujan problem would require. For problem 398, which asks whether n! + 1 = x^2 has only the solutions n = 4, 5, 7, this paper does not settle it but restricts the search to x±1 that are neither 7-free nor prime powers.

Source: https://math.colgate.edu/~integers/vol25.html.

Bears on. #398

Results to transcribe.

• Theorem 1(i): For any k >= 2 there are finitely many solutions of n!+1 = x^2 with x±1 k-free; for k = 7 the only candidates are (4,5), (5,11), (7,71).
• Theorem 1(ii): For any l >= 2 there are finitely many solutions with x±1 having fewer than l prime divisors; when x±1 is a prime power the only candidate is (4,5).
• Theorem 2 (Section 4): For k, l >= 2 and an integer polynomial P in m variables, only finitely many solutions have x = x_1...x_m P(x_1,...,x_m) ± 1 with every x_i a product yz, y k-free and omega(z) < l. A closing remark asks whether a suitable P makes this set contain all but a few odd positive integers, which would resolve the Brocard-Ramanujan problem.
• Toolkit: Uses Chebyshev's bounds on pi(n), Stirling's estimate for n!, and Legendre's formula for nu_p(n!) applied to the factorization n! = (x-1)(x+1).