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Statements

Throughout, (1) is the equation $1=1/x_1+\cdots+1/x_n$ and (3) is the condition $0<x_1<x_2<\cdots<x_n$ on its integer solutions. Printed p. 194 (PDF p. 3) states the following, after quoting Nakayama's theorems on $N(a,b)=2$ and $N(3,b)=3$.

• Kürschák's theorem and gaps of size $2$. The sum of the reciprocals of consecutive integers is never an integer (Kürschák, Math. és Phys. Lapok 27 (1918), cited in footnote 4 with Pólya--Szegő and Obláth), so every solution of (1) with (3) has some gap $x_{i+1}-x_i\ge2$.
• Conjecture (gap $3$). Experience shows that there is always a gap $x_{i+1}-x_i\ge3$; Erdős writes that he has not been able to prove this conjecture. The solution $2,3,6$ for $n=3$, whose gaps are $1$ and $3$, shows that the bound $3$ cannot be raised in general.
• Possible strengthening (large gaps). It is possible that for every positive $c$ there is $n_0$ such that for $n>n_0$ every solution of (1) with (3) has some gap $x_{i+1}-x_i>c$.
• Conjecture (ratio $3$). For every solution of (1) with (3), $x_n/x_1\ge3$, with equality only for $n=3$, $x_1=2$, $x_2=3$, $x_3=6$.
• Probable statement (unbounded ratio). Probably $x_n/x_1\to\infty$: for every fixed $q>0$ there is $n_0$ such that for $n>n_0$ every solution of (1) with (3) has $x_n>qx_1$.
• Question (counting). Erdős introduces this item as one of some interesting, as yet unsolved problems concerning the solutions of (1). For given $n$ let $f_1(n)$ be the number of solutions of (1) in positive integers and $f_2(n)$ the number of solutions satisfying (3); he asks to give the functions $f_1(n)$ and $f_2(n)$, or to determine functions asymptotically equal to them. The paragraph contains no assessment of the question's difficulty.

Source. Erdős, Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól, Mat. Lapok 1 (1950), 192--210; printed p. 194 (PDF p. 3), with footnote 4. Read on the page image (Hungarian; the OCR layer garbles formulas); the English summary on p. 210 does not restate these items.

Read depth. Claims checked: the six items were read clause by clause on the page image. They are conjectures and questions; nothing is proved on this page.

Later standing

• The gap-$3$ conjecture is the question of Problem 287, which the site cites to Erdős's 1932 paper on Kürschák's theorem and to the 1980 monograph; this page records only that the 1950 paper states it.
• The unbounded-ratio statement is contradicted by Croot's short-intervals theorem (Main Theorem): for every $N>1$ there is a representation $1=\sum1/x_i$ with $N<x_1<\cdots<x_k\le(e+O(\log\log N/\log N))N$, and such a representation has more than $N$ terms because each term is below $1/N$; so solutions with arbitrarily many terms and $x_k/x_1<3$ exist. Erdős and Graham wrote in 1980 (p. 34) the opposite expectation, that the least ratio $x_n/x_1$ over $n$-term representations "seems likely" to tend to $e$. Croot's theorem gives such ratios below $e+o(1)$ for infinitely many $n$; this settles the limit-inferior form in which Croot's preprint restates the question, not the limit over every $n$.
• The same representations refute the ratio-$3$ conjecture: for large $N$ they have $x_k/x_1<e+O(\log\log N/\log N)<3$. The large-gap strengthening is not assessed here.
• The counting question for $f_2(n)$ is Problem 148 (the number $F(k)$ of $k$-term representations of $1$ by distinct unit fractions).

Dependencies

None.

Bears on

• Problem 287: the gap-$3$ conjecture and its large-gap strengthening, as posed in 1950.
• Problem 148: the question on $f_2(n)$.