Luca 2002 diophantine equation result m

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proposition_1: Luca's proposition that the abc conjecture implies that, for every integer polynomial P of degree d at least 2, the equation P(x) = n!, with x an integer, has only finitely many solutions (x, n).

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Luca, Florian, The {D}iophantine equation {$P(x)=n!$} and a result of {M}. {O}verholt. Glas. Mat. Ser. III 37(57)(2) (2002), 269--273. No notice is printed in the copy read, the journal's PDF (its first page carries only the header "GLASNIK MATEMATIČKI Vol. 37(57)(2002), 269 – 273"); the journal's article page states no copyright or license term (https://web.math.pmf.unizg.hr/glasnik/vol_37/no2_04.html, read 2026-10-02), and its home page names the publishers, offers free access to volumes 33--57 and states no copyright, license or Creative Commons terms (https://web.math.pmf.unizg.hr/glasnik/, read 2026-10-02), free access without a named license being no license; the term is unstated.

For P in Z[X] of degree d >= 2, the paper studies the equation P(x) = n! with x an integer. Proposition 1 (p. 270) shows that the abc conjecture, in the form max(|A|, |B|, |C|) < C(eps) N(ABC)^(1+eps) for coprime nonzero A + B = C with N the radical, implies that the equation has only finitely many solutions (x, n); the surrounding text and the abstract state these as integer solutions with n > 0. This generalizes Overholt's result that a weak form of abc gives finitely many solutions of the Brocard-Ramanujan equation x^2 - 1 = n!. The proof reduces to a monic equation without a degree d - 1 term, disposes of the pure power case z^d = c n! by a prime in (n/2, n), and otherwise applies abc with eps = 1/(2d) to a three-term equation, bounding the radical of n! by 4^n; the resulting bound log|z| << n, with Stirling's formula, bounds n. Read status: claims checked for Proposition 1; the proof was read for structure only.

Source: https://web.math.pmf.unizg.hr/glasnik/vol_37/no2_04.html.

Bears on. #393: under the abc conjecture, Proposition 1 (p. 270) applied to the finitely many polynomials of degree at least 2 that the problem page's reduction attaches to each value m gives finitely many n with f(n) = m, so f(n) tends to infinity; the reduction is the problem page's, the paper names no f, and the result is conditional on abc (proposition_1), #398: the case P(X) = X^2 - 1 of Proposition 1 gives, under the abc conjecture, finitely many n with n! = x^2 - 1; it does not show that n = 4, 5, 7 are the only ones (proposition_1).

Results. Page numbers are those printed in the journal, pp. 269--273.

• Proposition 1 (p. 270): the abc conjecture implies that P(x) = n!, for P in Z[X] of degree d >= 2, has only finitely many solutions (x, n).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.