Erdos 1937 uber diophantische gleichungen der form und

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equation_ia: The observation in Erdős and Obláth's introduction that n! = x^2 + y^2 has no solution in positive integers for n >= 7 and for n = 3, 4, 5, while 6! = 12^2 + 24^2, with no coprimality assumption.

satz_1: Erdős and Obláth's theorem that, apart from 2! = 1 + 1, no factorial is a sum or a difference of the pth powers of two coprime positive integers when p >= 3 is not a power of 2, with its corollary that n! + 1 and n! - 1 are not pth powers for n > 2.

satz_2: Erdős and Obláth's theorem that the difference of the eighth powers of two coprime integers is never a factorial, so that n! + 1 is never an eighth power.

satz_3: Erdős and Obláth's theorem, proved with the prime number theorem for the progressions 4k+1 and 4k+3, that for sufficiently large n the factorial n! is not a difference of the fourth powers of two coprime integers, with no threshold given.

satz_4: Erdős and Obláth's theorem, proved with the prime number theorem, that the equation n! ± m! = x^p with n > m > 1 and p > 1 has at most finitely many solutions.

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Erdős, P. and Obláth, R., Über diophantische Gleichungen der Form $n!=x^p\pm y^p$ und $n!\pm m!=x^p$. Acta Litt. ac Sci. Reg. Univ. Hung. Fr.-Jos., Sect. Sci. Math. 8 (1937), 241-255.

The German paper (read as a scan) studies n! = x^p + y^p (p > 1) and n! = x^p - y^p (p > 2). The case p = 2 of the first equation is settled in the introduction with no coprimality assumption: for n at least 7 a prime q = 4k+3 between n/2 and n divides n! exactly once, which rules out n! = x^2 + y^2, and 3 | n!, 9 ∤ n! rules out n = 3, 4, 5; but 6! = 12^2 + 24^2 (pp. 241-242). From there on x and y are coprime, and Satz 1 (section 2, p. 250) proves that apart from the trivial x = y = 1, n = 2 neither equation has solutions when p >= 3 is not a power of 2 (the case of an odd prime p, from which every such p follows), with the corollary that n! ± 1 (n > 2) is not a p-th power. Section 3 handles p = 8, that is n! = x^8 - y^8, and Satz 2 (p. 251) states that the difference of the eighth powers of two coprime integers is never a factorial, from which the unsolvability for p = 2^alpha with alpha at least 3 follows; the case p = 4 needs sharp information on primes in the progressions 4k+1 and 4k+3 and is treated in sections 4 and 5 using the prime number theorem for arithmetic progressions, which gives only Satz 3 (p. 254): for sufficiently large n, n! is not a difference of the fourth powers of two coprime integers. Section 6 proves, again with the prime number theorem, Satz 4 (p. 255): n! ± m! = x^p with n > m > 1 and p > 1 has at most finitely many solutions. The main tool is formula (V) of section 1 (p. 247), an elementary upper bound for the contribution of the primes ak+1 to the prime factorization of n!, used in the cases a = 2p (Va) and a = 8 (Vb) and derived by a Chebyshev-de Polignac style transformation of Legendre's exponent formula and valid for all n from the start. Erdős and Obláth summarize the results as: factorials are in general not sums or differences of powers, and sums or differences of two factorials are in general not perfect powers. The paper lists the known solutions of n! ± m! = x^p, among them 5! + 4! = 12^2, and calls it probable that there are no others (p. 255).

Source: https://users.renyi.hu/~p_erdos/Erdos.html. No notice is printed in the file (no page carries a copyright or licence line); the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); a search of the publisher's repository (acta.bibl.u-szeged.hu, read 2026-10-02) locates the article at http://acta.bibl.u-szeged.hu/13485/ with no rights statement on the results page, the item page itself was not read, and no Crossref license is recorded; the term is unstated.

Read status. Claims checked: Satz 1 with its Korollar, Satz 2, Satz 3 with the Hilfssatz of section 4, Satz 4 and the observation on (Ia) were read clause by clause on the printed pages. The proofs were followed but not checked step by step.

Bears on. #399: the problem asks whether n! = x^k ± y^k has no solutions with xy > 1 and k > 2. For coprime x and y the paper excludes every k > 2 not a power of 2 (Satz 1), differences with k = 2^alpha, alpha at least 3 (Satz 2), and differences with k = 4 only for n beyond an unspecified threshold (Satz 3). Sums with k a power of 2 reduce, coprime or not, to (Ia), which leaves only n = 6, a case the paper does not discuss for k >= 4. The theorems for k > 2 assume x and y coprime and say nothing about x and y with a common factor.

Results. Equation (Ia) (pp. 241-242, unnumbered); Satz 1 and Korollar (p. 250); Satz 2 (p. 251); Satz 3 (p. 254), with the Hilfssatz of section 4 (p. 251) on its page; Satz 4 (p. 255). Formula (V) of section 1 (p. 247) is the proof tool of Satz 1 and Satz 2, summarized on their pages.

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