Erdos 1979 unconventional problems number theory math mag

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P. Erdős, Some unconventional problems in number theory, Math. Mag. 52 (1979), no. 2, 67--70; subtitle "A mélange of simply posed conjectures with frustratingly elusive solutions". A note in the paper says it grew out of Erdős's remarks at the fifth annual Mathematics and Statistics Conference held at Miami University in Oxford, Ohio, in October 1977.

Three 1979 papers share this title. This Math. Mag. paper (cited as [Er79] on the problem pages), the Astérisque 61 paper filed as erdos_1979_unconventional_problems_number_theory_asterisque ([Er79e]) and the Acta Math. Acad. Sci. Hungar. 33 paper filed as erdos_1979_unconventional_problems_number_theory ([Er79d]) are all called "Some unconventional problems in number theory"; their contents differ, though item 3 below and equation (8) of the Acta paper pose the same question.

The copy read for this card is a scan of the four printed pages 67--70 (PDF p. $n$ is printed p. $66+n$) with an OCR text layer (OmniPage 12) that garbles the formulas; all four pages were read on the page images. Provenance: a scan obtained in a survey download of September 2026 (its cache file name was 1979-22.pdf, the numbering of the Rényi Institute's Erdős archive); the download URL was not recorded; 809,204 bytes. Read status: claims checked for the statements listed below that the nine citing problems consume (read on the page images; item 2 re-read for Problems 677 and 678); the paper contains no proofs, and nothing stated in it was verified here. No notice is printed on the four scanned pages; the Crossref record for DOI 10.1080/0025570X.1979.11976756 (read 2026-10-02) names Informa UK Limited as publisher and no license, and the publisher's page could not be read on 2026-10-02 (tandfonline.com returned HTTP 403); the term is unstated.

Contents

The paper's twelve numbered items, with the statements the citing problems consume in full and the rest in brief.

1. Factorial powers (p. 67): with $f(n)=\sum1/p$ over $p<n$, $p\nmid\binom{2n}{n}$, the conjecture with Graham, Ruzsa and Straus [6] that $f(n)<C$; the conjecture that $\binom{2n}{n}$ is never squarefree for $n>4$, reduced to showing that $\binom{2^{k+1}}{2^k}$ is divisible by the square of an odd prime for $k\ge3$; the conjecture that $2^k$ is not a sum of distinct powers of $3$ for $k>8$; and the guess that $\binom{342}{171}$ is the largest $\binom{2n}{n}$ not divisible by the square of an odd prime.
2. Least common multiples (pp. 67--68): $M(n;k)=[n+1,\dots,n+k]$. Erdős conjectures $M(n;k)\ne M(m;k)$ for $m\ge n+k$ and, more generally, $M(n;k)\ne M(m;l)$ for $l\ge k$ and $m\ge n+k$, and expects $M(n;k)=M(m;l)$ to have very few solutions with $m\ge n+k$ and $l>1$, knowing only $M(4;3)=M(13;2)$ and $M(3;4)=M(19;2)$; for the products $A(n;k)=\prod_{i=1}^k(n+i)$ he conjectures that $A(n;k)=A(m;l)$ likewise has very few solutions with $m\ge n+k$ and $l>1$. "Suppose that $k\ge3$ and $m\ge n+k$. Observe that then, for each $k$, $M(n;k)>M(m;k)$ has infinitely many solutions. Yet I cannot decide whether the same is true for $M(n;k)>M(m;k+1)$. (The referee found two solutions, namely $M(96;7)>M(104;8)$ and $M(132;7)>M(139;8)$.)" With $n_k$ the smallest solution of $M(n;k)>M(m;k)$, $n_k/k\to\infty$ is "indeed easy" (added in proof) but no good upper bound is known; with $u_k$ the smallest integer with $M(u_k;k)>M(u_k+1;k)$, $u_k=(1+o(1))k$ and $u_k>k$, and Erdős guessed, and could not prove, that $M(t;k)\le M(T;k)$ for $t<u_k$ and $T>t$.
3. Unusual sieve processes (p. 68): the integers of the form (1) $n=ap^2+b$ with $a\ge1$, $0\le b<p$, $p$ prime. "It is easy to see by the sieve of Eratosthenes that almost all integers $n$ are of the form (1), but I could not prove that every sufficiently large integer is of this form. In fact, this seems rather unlikely." The variant (2) $n=ak^2+b$ with an integer $k\ge2$ was hoped to be solvable for every large $n$, but after a preliminary computer search Selfridge and Wagstaff think it quite possible that it fails for infinitely many $n$; with $g(x)$ and $G(x)$ the counts of exceptions to (1) and (2), the Brun--Selberg sieve gives $g(x)<c_1x(\log x)^{-c_2}$, and probably $G(x)<x^c$ for $x>x_0(c)$ for some $c>0$, perhaps for all $c>0$; generalizations (3) with sequences $u_i$, $v_i$.
4. Barriers (p. 68): $n$ is a barrier for $f$ if $m+f(m)\le n$ for all $m<n$. "Probably $V(m)$ has infinitely many barriers, but I am very far from being able to prove this. I cannot even prove that there is an $\epsilon>0$ for which $\epsilon V(m)$ has infinitely many barriers", where $V(m)$ is the number of distinct prime factors; the same for $\Omega(m)$ "is certainly unattackable by present day methods", and Selfridge found $99840$ to be the largest barrier for $\Omega(m)$ below $10^5$. For $d(n)$, since $\max(d(n-1)+n-1,d(n-2)+n-2)\ge n+2$, the most one can hope is (4) $\max_{m<n}(m+d(m))=n+2$ for infinitely many $n$: "It is extremely doubtful whether (4) has infinitely many solutions. In fact it is quite possible that $\lim_{n\to\infty}\max_{m<n}(m+d(m)-n)=\infty$." Erdős and Selfridge found that $n=24$ satisfies (4) and were persuaded that any larger solution would be far too large for their computations to find. The product $F(m)=\prod\alpha_i$ of the exponents has infinitely many barriers.
5. Translation properties (p. 69): squarefree numbers and sequences avoiding multiples of pairwise coprime $b_i$ with $\sum1/b_i<\infty$ have the translation property, and by Brun's method so do the sequences avoiding multiples of such $b_i$ when $\sum_{b_i<x}1/b_i=o(\log\log x)$; questions for sums of two squares and for the integers composed of the primes of one class, when the primes are split into two classes each with more than $cx/\log x$ members up to $x$ (the primes from any point on never have the property); the least shift $t_n$ for squarefree numbers is expected to exceed $\exp n^c$.
6. Consecutive primes and squarefree numbers (p. 69): Cramér's conjecture; then, for consecutive squarefree numbers $Q_n$, the best upper bound known to Erdős is that of Richert and Rankin [10, 11], $Q_{n+1}-Q_n<n^{2/9+\epsilon}$ for every $\epsilon>0$ and $n>n_0$. He says there is "no doubt" that the exponent $2/9+\epsilon$ can be replaced by $\epsilon$, with no proof in sight, and that $Q_{n+1}-Q_n<c\log n$ may hold, though he is "very doubtful" of it. In the other direction he calls it easy that $\limsup_{n\to\infty}(Q_{n+1}-Q_n)\log\log n(\log n)^{-1}\ge\pi^2/6$, and knows of no improvement of it. Erdős proved [5] that $\lim(1/x)\sum_{Q_n<x}(Q_{n+1}-Q_n)^\alpha=c_\alpha$ for $0\le\alpha\le2$ (5), Hooley [8] extended it to $\alpha\le3$, and it should hold for every $\alpha$.
7. Divisors (p. 69): "The density of integers $n$ which have two divisors $d_1<d_2<(1+\epsilon)d_1$ is $1$ for every $\epsilon>0$. I can prove that the density exists, but cannot prove that it is $1$, even for large values of $\epsilon$." The stronger conjecture: with $d^+(n)$ the number of $k$ for which $n$ has a divisor $t$ with $2^k<t<2^{k+1}$, $d^+(n)/d(n)\to0$ for almost all $n$. Erdős refers to a long paper with R. R. Hall on such problems.
8. Sums of divisors of $n!$ (p. 70): $h(n)$, the least $h$ such that every $\mu$ with $1\le\mu<n!$ is a sum of at most $h$ distinct divisors of $n!$; $h(n)\le n$ by an easy induction; conjectures $h(n)=o(n)$, $h(n)=o(n^\epsilon)$ and hopefully $h(n)<(\log n)^c$.
9. The Erdős--Straus conjecture $4/n=1/x_1+1/x_2+1/x_3$ (p. 70).
10. The equation $x^xy^y=z^z$ (p. 70): "Forty years ago I asked: does $x^xy^y=z^z$ have any nontrivial solutions in integers? Chao Ko found infinitely many solutions [1]; perhaps he found them all."
11. Consecutive prime gaps (p. 70): with Turán, $d_{k+1}>d_k$ and $d_{k+1}<d_k$ each happen infinitely often; whether $d_{k+2}>d_{k+1}>d_k$ or its reverse happens infinitely often is open, with a prize offered.
12. Gaps between totatives (p. 70): the conjecture (6) $\sum_{i<\phi(n)}(r_{i+1}-r_i)^2<Cn^2/\phi(n)$ over the integers $r_i$ prime to $n$, with a prize offered; Hooley proved the version with exponent $\alpha<2$.

Compiled scope

All four pages were read on the page images and the statements above were checked there. The paper proves nothing, and nothing it states was verified here or independently reviewed.

Bears on. #144: item 7 states the problem's conjecture in the stronger form $d_1<d_2<(1+\epsilon)d_1$ for every $\epsilon>0$, notes that the density exists, and states the stronger conjecture $d^+(n)/d(n)\to0$; #675: item 5 (p. 69, PDF p. 3, page image) asks whether the sums of two squares, and the integers composed of one class of a split of the primes into two classes each with more than $cx/\log x$ members up to $x$, have the translation property, and expects the least shift $t_n$ for the squarefree numbers to exceed $\exp n^c$, after stating that the squarefree numbers and the integers avoiding multiples of pairwise coprime $b_i$ with $\sum1/b_i<\infty$ have it, and that by Brun's method the weaker condition $\sum_{b_i<x}1/b_i=o(\log\log x)$ suffices; #413: item 4 poses barriers for $V(m)$, the number of distinct prime factors, and for $\epsilon V(m)$, the problem's two questions; #647: item 4, equation (4), with the remark that $24$ satisfies it and any further solution must be enormously large, is the problem's question; #674: item 10 poses $x^xy^y=z^z$ and reports Chao Ko's infinitely many solutions [1]; #676: item 3, equation (1), with the sieve remark that almost all $n$ have the form and the doubt that all large $n$ do; #208: item 6 states the Richert--Rankin bound $n^{2/9+\epsilon}$ for gaps between squarefree numbers, the expectation of $n^\epsilon$, the doubt about $c\log n$, and the $\limsup$ bound with $\pi^2/6$, "never been improved": the first question is the $n^\epsilon$ expectation, and the second, an upper bound matching the $\pi^2/6$ lower bound, is not posed in the item; #677: item 2 (p. 67, PDF p. 1, page image) states the problem's conjecture $M(n;k)\ne M(m;k)$ for $m\ge n+k$, its generalization $M(n;k)\ne M(m;l)$ for $l\ge k$, and the two known solutions $M(4;3)=M(13;2)$ and $M(3;4)=M(19;2)$ of the general equation; #678: item 2 (p. 67, PDF p. 1, page image) poses $M(n;k)>M(m;k+1)$ with the referee's two solutions, the problem's question, in a sentence that leaves open whether $k$ is fixed; p. 68 (PDF p. 2) carries the $n_k$ and $u_k$ remarks the site's commentary repeats.

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