Erdos 1975 product consecutive integers is never power

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conjecture_p292: Erdős and Selfridge's unproved strengthening of their Theorem 2: for k at least 4 and n + k at least p^(k), some prime greater than k divides (n+1)...(n+k) to the first power; the statement that would settle Problem 137 for k at least 4.

remark_p300: Erdős and Selfridge's unproved Section 4 assertions: for each positive d a threshold t_d beyond which (n+d)(n+2d)...(n+td) is never a perfect power, and finiteness of solutions of a gapped product equation for fixed t; the starting point for Problem 672.

theorem_1: Erdős and Selfridge's theorem that (n+1)(n+2)...(n+k) = x^l has no solution with k at least 2, l at least 2 and n at least 0, the case of one interval in Problem 930 and of common difference one in Problem 672.

theorem_2: Erdős and Selfridge's stronger result: for k at least 3, l at least 2 and n + k at least the least prime p^(k) that is at least k, some prime p at least k divides (n+1)...(n+k) to an exponent not divisible by l.

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P. Erdős, J. L. Selfridge: The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), no. 2, 292--301 (MR 51 #12692; Zentralblatt 295.10017).

Theorem 1 settles the old conjecture that (n+1)...(n+k) = x^l has no solution with k at least 2, l at least 2 and n at least 0, that is, no run of at least two consecutive positive integers multiplies to a perfect power. The proof goes through the stronger Theorem 2: if k is at least 3, l at least 2 and n + k is at least p^{(k)} (the least prime that is at least k), then some prime p at least k divides (n+1)...(n+k) to a power not divisible by l. The method is elementary but intricate: assuming Theorem 2 fails, write n + i = a_i x_i^l with a_i l-th-power free and all prime factors of a_i below k, prove that for each l' < l the products of l' of the a_i (repetition allowed) are distinct (Lemma 1), and derive size contradictions by counting the a_i: for l > 2 through a graph-theoretic bound (Lemma 3) when k >= 30000 and counts of a's with small prime factors below that (Section 2), and for l = 2 through estimates for the powers of 2 and 3 and for the product of the primes below k when k >= 71, finishing k < 71 by a hand analysis of a's with only small prime factors (Section 3). Lemma 1 (p. 293) assumes, as Section 1 does, that Theorem 2 fails for some k, l, n with n > k, so that n > k^l by (2); its proof shows that the ratio of two different such products of l' of the a_i is never the l-th power of a rational. Between the theorems and Section 1 (pp. 292-293) the paper conjectures, without proof and calling it very deep, a strengthening of Theorem 2: for k at least 4 and n + k at least p^{(k)}, some prime greater than k divides (n+1)...(n+k) to the first power. Section 4 (pp. 300-301) states on p. 300 (PDF p. 9) an unnumbered arithmetic-progression assertion: for each fixed positive $d$ there is a $d$-dependent $t_d$ such that $(n+d)(n+2d)\cdots(n+td)$ is never a perfect power for $t>t_d$. The paper offers it as an expectation ("there must be" such a $t_d$), with no proof and no threshold uniform in $d$. The same section says the authors' methods can prove finiteness for fixed $t$ in the equation $(n+d_1)\cdots(n+d_k)=x^\ell$ with $1=d_1<\cdots<d_k\leq k+t$, without giving a proof. Theorem 1 is the case of one interval in Problem 930 on products over several disjoint intervals and the case $d=1$ of Problem 672 on powers in products of arithmetic progressions, while the Section 4 remarks are the starting point for the rest of Problem 672. Theorem 1 does not decide Problem 137 on powerful products of consecutive integers; the unproved strengthening above is the statement that would.

Source: https://users.renyi.hu/~p_erdos/1975-46.pdf. No notice is printed on the hosting archive's scan; the article's Project Euclid page (https://doi.org/10.1215/ijm/1256050816) could not be read on 2026-10-02 (it served a bot challenge), its archived capture of 2025-08-21 shows no article-level copyright line and only the footer "© 2024 Project Euclid, a Duke University Press initiative", and the Crossref record names Duke University Press as publisher and no license; the publisher's own site (read 2026-10-02) prints "© 2024 Duke University Press. All Rights Reserved.", every other right reserved.

Bears on. #137: the conjectured strengthening of Theorem 2 (pp. 292-293), if true, would, with the paper's remark on n at most k (p. 292), give a negative answer for every k at least 4; it is unproved, Theorem 2 itself (a prime at least k to a power that is odd when l = 2) does not exclude a powerful product, and nothing here settles the problem. #672: Theorem 1 is the case of common difference $d=1$; the Section 4 threshold $t_d$ for other $d$ is asserted without proof and leaves the problem open. #930: Theorem 1 is the case $r=1$, with length $k=2$; it says nothing about two or more intervals.

Results.

• Theorem 1 (p. 292): a product of two or more consecutive positive integers is never a power.
• Theorem 2 (p. 292): for k at least 3, l at least 2 and n + k at least p^{(k)}, a prime at least k divides the product to a power not divisible by l.
• Conjecture (pp. 292-293): the unproved strengthening of Theorem 2, a prime greater than k to the first power.
• Section 4 remarks (p. 300): the unproved threshold $t_d$ for arithmetic progressions and the finiteness assertion for products with gaps.

Sampled scope. Theorem 1 is on printed p. 292 (PDF p. 1). All ten printed pages, 292--301, were read on the page images of the Rényi scan, and the statements, the proof outline and the Section 4 remarks above were checked against them; no proof was verified, and no new proof credit is given.

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