Erdos 1932 egy kurschak fele elemi

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theorem_1: States that 1/a + 1/(a+d) + ... + 1/(a+nd) is not an integer for any positive integers a, d, n, the generalization of Kürschák's theorem on consecutive integers, with the prime-power lemma (2) behind the proof.

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Erdős, P., Egy Kürschák-féle elemi számelméleti tétel általánosítása [Generalization of an elementary number-theoretic theorem of Kürschák]. Matematikai és Fizikai Lapok 39 (1932); eight-page offprint. The site's reference key [Er32] prints the title as "Egy Kürschak-féle elemi számelméti tétel áltadánositása" in "MAt. es Phys. Lapok".

The copy read for this card is a scan of the offprint ("Különlenyomat a «Matematikai és Fizikai Lapok» XXXIX. kötetéből. Budapest, 1932."): pp. 1--7 carry the Hungarian text and p. 8 a German summary ("Verallgemeinerung eines elementar-zahlentheoretischen Satzes von Kürschák"). Its OCR layer garbles the formulas; the statements below were read on the rendered page images. Read status: the theorem (1) and the lemma (2) were read clause by clause (claims checked); the proof (pp. 2--7) was read for structure and is sketched on the result page; no proof was rewritten and none has been independently reviewed. No copyright or license line is printed (pp. 1--2 and 7--8 read; besides the Hungarian offprint line above, the offprint prints only "Sonderabdruck aus «Matematikai és Fizikai Lapok» Band XXXIX. Budapest 1932."); the hosting archive's site footer speaks for the site, not the paper ("(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", https://users.renyi.hu/~p_erdos/, read 2026-10-02); no publisher page exists for this edition, so none was consulted, and no Crossref license is recorded; the term is unstated.

This short Hungarian note (read as a scan) generalizes Kürschák's elementary theorem that a partial sum of the harmonic series 1/k + ... + 1/n is never an integer. The main theorem, displayed as (1) on page 1, states that for arbitrary positive integers a, d, n the sum 1/a + 1/(a+d) + ... + 1/(a+nd) cannot be an integer, so the harmonic-progression case a = k, d = 1 is recovered. The proof first reduces to gcd(a,d) = 1, then rests on an auxiliary lemma that one of a+d, a+2d, ..., a+nd is divisible by a prime power p^alpha exceeding n; clearing denominators, the term with that factor removed carries a strictly lower power of p than the numerator's other terms and the common denominator, so the quotient (4) cannot be an integer. The lemma is used for d at least 4; the remaining cases d = 1, 2, 3 are handled separately: d = 1 is Kürschák's theorem, d = 3 (and any odd d) goes by the highest power of 2 dividing a term, and d = 2 by the highest power of 3. Erdős cites Theisinger (1915), Obláth (1918) and Kürschák (1918) as the earlier theorems being generalized. For Problem 287, which asks whether distinct denominators with unit fractions summing to 1 must contain a gap of at least 3, the site cites this paper for the bound 2: the reciprocals of an arithmetic progression never sum to an integer, so in particular no run of consecutive integers can represent 1. For Problem 288, on pairs of intervals, the case d = 1 is the background fact that no interval of two or more consecutive integers has an integer reciprocal sum.

Source: https://users.renyi.hu/~p_erdos/Erdos.html.

Bears on. #287 (the case d = 1 gives the gap-at-least-2 fact; the theorem covers complete arithmetic progressions only; beyond excluding the all-gaps-2 pattern through its case d = 2, it says nothing about the gap-3 question); #288 (the case d = 1 shows that the reciprocal sum over one interval of two or more consecutive integers is never an integer; beyond two adjacent intervals, whose union is a single interval, the theorem says nothing about sums over two intervals, which the problem asks about).

Results.

• Theorem (1), p. 1: For any positive integers a, d, n, the sum 1/a + 1/(a+d) + ... + 1/(a+nd) is never an integer.
• Auxiliary lemma (2), p. 2: for gcd(a,d) = 1 and d at least 4, some term of a+d, a+2d, ..., a+nd is divisible by a prime power p^alpha greater than n, which drives the non-integrality proof.

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