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Statement

$N(a,b)$ is the least $n$ for which $a/b=1/x_1+\cdots+1/x_n$ has a solution in integers $0<x_1<\cdots<x_n$. Printed p. 195 (PDF p. 4) opens the discussion of $N(a,b)$ for fixed $a$: for $a=2$ and $a=3$ the maximum over $b$ is $2$ and $3$ (by display (4), $N(a,b)\le a$; for example $3/7$ needs three terms), and for $a=4$ the paper already finds the maximum of $N(a,b)$ seriously hard to determine:

STRAUSS-szal együtt az a sejtésünk, hogy $N(4,b)\le3$, ha $b>4$. STRAUSS $4<b<5000$ esetére be is bizonyította ezt a sejtést.

(p. 195; in a translation made here: together with Straus, Erdős conjectures that $N(4,b)\le3$ for $b>4$, and Straus has proved this for $4<b<5000$.) The English summary on printed p. 210 (PDF p. 19) states it as "Strauss and the author conjecture that $N(4,b)<4$ for every $b\ge4$, Strauss proved this for $b<5000$."

Source. Erdős, Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól, Mat. Lapok 1 (1950), 192--210; printed p. 195 (PDF p. 4), first paragraph, and the English summary on printed p. 210 (PDF p. 19). Read on the page images (the OCR layer garbles the formulas; the two passages were located through it and read on the images).

Read depth. Claims checked: the two passages were read clause by clause on the page images. Nothing is proved on this page; the paper gives no argument for the conjecture and none for Straus's verification.

Relation to Problem 242

The site's formulation asks for exactly three distinct denominators $1\le x<y<z$ with $4/n=1/x+1/y+1/z$ for every $n>2$. Since $N(a,b)$ is defined for $0<a<b$, the 1950 statement covers $b>4$; the site's cases $n=3$ and $n=4$ are $4/3=1+1/4+1/12$ and $4/4=1/2+1/3+1/6$ (checked here). "At most three" and "exactly three" agree for $b>4$: a representation with one or two distinct terms becomes one with three distinct terms by splitting the term with the largest denominator by $1/y=1/(y+1)+1/(y(y+1))$, once or twice (checked here; an elementary remark, not the paper's). The site cites the paper as [Er50c] and dates the conjecture's first appearance in print to Obláth's paper, submitted in 1948; the 1950 paper's own text credits the conjecture to Erdős and Straus jointly and reports Straus's finite check.

Dependencies

None.

Bears on

• Problem 242: the earliest statement of the Erdős--Straus conjecture in the library, with the range $4<b<5000$ verified by Straus as of 1950.