Schur 1916 uber die kongruenz

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dicksonscher_satz_p115: The paper's main application: the Hilfssatz on difference-free partitions gives Dickson's theorem on the Fermat congruence with the explicit bound M = m! e + 1, for every m, not only for prime m.

hilfssatz_p114: Schur's lemma, the finiteness of the Schur numbers with the bound m! e, proved by iterated differences; the paper says nothing about graphs or Ramsey numbers.

lower_bound_p117: Schur's tripling construction for difference-free partitions of an initial interval, giving the exponential lower bound (3^m - 1)/2 on the Schur numbers; the paper says nothing about graphs or Ramsey numbers.

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I. Schur, Über die Kongruenz $x^m+y^m\equiv z^m\pmod p$, Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1916), 114--117.

The copy read for this card is assembled from the Göttingen digitization (GDZ) of the volume, five PDF pages: PDF p. 1 is the GDZ terms-of-use cover of the article download (structure LOG_0014), PDF pp. 2--4 are printed pp. 114--116 from that download, and PDF p. 5 is printed p. 117, the volume's page image 00000124 (structure PHYS_0125), which the article download omits because GDZ's structure map assigns that page to the following article (Mehmke's, which begins on it); printed p. $n$ is PDF p. $n-112$. The article pages carry no text layer; everything below was read on the page images. The text of p. 116 ends "Genügt nun das Schema ..." and p. 117 begins "dieser Bedingung, so liefern ...", so the article is complete. Source: https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0025. Provenance: PDF pp. 1--4 are the GDZ article download obtained on 2026-09-04; PDF p. 5 was retrieved from https://gdz.sub.uni-goettingen.de/content/PPN37721857X_0025/1000/0/00000124.jpg (HTTP 200, one request; the page's identity and printed number from the volume's METS file, one request) and wrapped as a PDF page without recompression; the assembled copy is 643,292 bytes. Its first page is the digitizing library's terms sheet, which states that "The Goettingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes" and that "Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library"; the article pages are image-only and print no notice, every other right reserved.

Read status: claims checked for the Hilfssatz, the conclusion $M=m!\,e+1$ and the lower bound $N_m\ge(3^m-1)/2$ of p. 117 (read clause by clause on the page images); the proof of the Hilfssatz (pp. 115--116) and the p. 117 construction were read in full on the page images and are not independently reviewed.

Contents

• Introduction (p. 114): Dickson's theorem, that the congruence (1) $x^m+y^m\equiv z^m\pmod p$ has a solution in integers coprime to $p$ once the prime $p$ exceeds a bound $M$ depending only on $m$; Dickson's two proofs rest on rather involved computations, and Schur shows the theorem follows almost at once from a very simple lemma that he places closer to combinatorics than to number theory.
• Hilfssatz (p. 114): if the numbers $1,2,\ldots,N$ are distributed in any way into $m$ rows and $N>m!\,e$, then some row contains two numbers whose difference lies in the same row ($e$ the base of the natural logarithms, footnote 3). This is Schur's theorem: the Schur number $S(m)$ is below $m!\,e$.
• Deduction of Dickson's theorem (pp. 114--115): with $g$ a primitive root modulo $p$ and $p-1=mq$, the residues of $g^\nu$ are distributed into $m$ rows by $\nu\bmod m$; for $p-1>m!\,e$ the Hilfssatz gives indices with $r_{\mu+\gamma m}-r_{\mu+\beta m}=r_{\mu+\alpha m}$, so $x=g^\alpha$, $y=g^\beta$, $z=g^\gamma$ solve (1); when $m\nmid p-1$ the same holds with $d=\gcd(m,p-1)$. Conclusion, p. 115: "Der Dicksonsche Satz ist also richtig, wenn $M$ gleich $m!\,e+1$ gesetzt wird."
• Proof of the Hilfssatz (pp. 115--116): pick the fullest row, pass to its differences from the least element, iterate; the counts satisfy $n_\mu-1\le n_{\mu+1}(m-\mu)$ (display (5)), which sums to $n_1/(m-1)!<e$ and $N\le mn_1<m!\,e$.
• Remark (p. 116): Dickson had shown by cyclotomy, for prime $m$ only, that $M=m^4-6m^3+13m^2-6m+1$ suffices; such a bound cannot be reached by the present method alone, and getting the smallest $M$ this way amounts to determining, for given $m$, the largest $N_m$ such that $1,\ldots,N_m$ can be distributed into $m$ rows with no row containing the difference of two of its numbers.
• The lower bound (pp. 116--117): if the scheme $x_1,x_2,\ldots;\ \ldots;
  u_1,u_2,\ldots$ of $m$ rows for the numbers $1,\ldots,N_m$ satisfies the condition, then the $m+1$ rows $3x_1,3x_1-1,3x_2,3x_2-1,\ldots$; $\ldots$; $3u_1,3u_1-1,3u_2,3u_2-1,\ldots$; $1,4,7,\ldots,3N_m+1$ form a scheme for $3N_m+1$ ("wie man leicht erkennt"), illustrated for $m=2$ by the rows $1,4$ / $2,3$ passing to $3,2,12,11$ / $6,5,9,8$ / $1,4,7,10,13$. Hence $N_{m+1}\ge3N_m+1$, and with $N_1=1$ the number $N_m$, shown earlier to be below $m!\,e$, is at least $1+3+3^2+\cdots+3^{m-1}=(3^m-1)/2$; this is of higher order than Dickson's bound and exceeds it already for $m\ge7$. Footnote 1: $N_m$ equals $(3^m-1)/2$ exactly only for $m\le3$. The article ends here; Mehmke's paper begins on the same page.

Compiled scope

Printed pp. 114--117 were read on the page images. No statement here is independently reviewed.

Bears on. #554: the site credits Schur with $C^k\ll R_k(K_3)\ll k!$; the paper contains the Hilfssatz, the source of the factorial upper bound on Schur numbers, and on p. 117 the construction $N_{m+1}\ge3N_m+1$ giving the exponential lower bound $N_m\ge(3^m-1)/2$ on the largest sum-free-partitionable interval (lower bound, p. 117), both read on the page images; it says nothing about graphs or Ramsey numbers. Only the lower bound passes to $R_k(K_3)$ by a later translation, the difference coloring that gives $R_k(K_3)\ge S(k)+2$; the factorial upper bound on $R_k(K_3)$ comes from running the Hilfssatz's argument on edge colorings (the Greenwood--Gleason recursion), not from $S(k)<k!\,e$. #483: the Hilfssatz is the origin of the factorial upper bound on the Schur function, $f(m)\le\lfloor m!\,e\rfloor+1$ in the site's convention $f(m)=S(m)+1$ (read on the page image of p. 114 on 2026-09-18; the Hilfssatz page records the statement), and p. 117 gives the lower bound $S(m)=N_m\ge(3^m-1)/2$, so $f(m)\ge(3^m+1)/2$ (lower bound, p. 117); the exponential upper-bound question $f(k)<c^k$ that Problem 483 asks is not touched.

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