Guy 2004 unsolved problems number theory

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Richard K. Guy, Unsolved Problems in Number Theory, third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. The copyright page (PDF p. 5) prints the Library of Congress record "Unsolved problems in number theory / Richard Guy.--[3rd ed.]", Mathematics Subject Classification (2000) 00A07, 11-01, 11-02, ISBN 978-1-4419-1928-1, ISBN 978-0-387-26677-0 (eBook), DOI 10.1007/978-0-387-26677-0, "© 2004 Springer Science+Business Media New York", "Originally published by Springer-Verlag New York, LLC in 2004" and "Softcover reprint of the hardcover 3rd edition 2004"; the author at the Department of Mathematics and Statistics, University of Calgary. The preface to the third edition (pp. v--vi) is dated Calgary 2003-09-16, says that sections A20, C21, D29 and F32 are new and that lists of OEIS sequence numbers now end about half of the sections, and remembers Erdős: "the various monetary rewards he offered may still be negotiable via Ron Graham". Cited as [Gu04] on the problem pages, the site's key; most pages carry only the reference entry, and thirteen name a section (A2, B2, B14, B26, B31, B33, B35, C2, C15, C16, E2, E5, E16). The introduction (pp. 1--2) lists the book's sources of problems, among them the Erdős and Graham monograph filed as erdos_1980_old_new_problems_results_combinatorial_number_theory, and fixes the conventions: "number" means natural number, $c$ is an absolute positive constant not necessarily the same at each appearance, and $m\perp n$ means $\gcd(m,n)=1$.

The copy read for this card is the whole eBook, one file because the eBook is issued as one volume: 455 PDF pages, the publisher's 2012 scan of the printed pages (Acrobat 8.0 and Distiller 9.0 in the file's metadata, a September 2012 creation date, PDF/X-3 subtype) with a text layer that reads the prose and garbles the symbols ($\sigma$ as an O followed by a double quotation mark, $\phi$ as ¢, Turán as Thran, section labels with l for 1) and drops most displayed formulas, so the displays quoted below were read on the page images. Page mapping: PDF p. 1 is the front cover (author, title, edition and publisher), with no text layer; the roman front matter pp. i--xviii (series page, title, copyright, the three prefaces, glossary of symbols, contents) is PDF pp. 2--19 (roman p. $n$ is PDF p. $n+1$); printed pp. 1--69 are PDF pp. 20--88 (printed p. $n$ is PDF p. $n+19$); the blank verso p. 70 that ends part A is not in the file, so printed pp. 71--427 are PDF pp. 89--445 (printed p. $n$ is PDF p. $n+18$); the blank p. 428 after the author index is also absent, so the General Index pp. 429--437 is PDF pp. 446--454 (printed p. $n$ is PDF p. $n+17$); PDF p. 455 is the series list. The edition is the publisher's DRM-free PDF of the whole eBook, https://doi.org/10.1007/978-0-387-26677-0 (the book's DOI, resolving to the publisher's platform), 46,813,794 bytes. The file prints "© 2004 Springer Science+Business Media New York" and "All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis" on its copyright page (PDF p. 5), every other right reserved.

Read status: claims checked, for the passages of the twelve sections that problem pages name, each read clause by clause on the page images of PDF pp. 98, 122, 143, 147, 152, 155--156, 182, 211, 216, 330, 333 and 348 (printed pp. 80, 104, 125, 129, 134, 137--138, 164, 193, 198, 312, 315 and 330); the passages for the other 82 citing pages were located here by searching the text layer and read there, clause by clause where a row restates them, so those rows rest on the text layer and on the sections' printed headings. The front matter, the introduction and the contents were read in the text layer for identification and for the page mapping. The book is a problem collection that reports results without proof, apart from a few short arguments such as the identity in B31 (p. 131); each section ends in its reference list, and not every statement is supported by a reference (the General Index, p. 429, lists the names whose mention is unsupported by references). Its statements of what is open describe the state when it went to press: the preface is dated 2003-09-16, and the text records events into April 2004, among them the STOP PRESS of 04-04-09 in A5. Nothing here is independently reviewed.

Contents

The introduction (p. 2, PDF p. 21) partitions the book "somewhat arbitrarily at times" into six parts, each a run of sections labeled by the part's letter; a section is a short problem essay ending in its reference list and, in this edition, often an OEIS line. Introduction (pp. 1--2, PDF pp. 20--21); A. Prime Numbers, A1--A20 (pp. 3--69, PDF pp. 22--88); B. Divisibility, B1--B50 (pp. 71--158, PDF pp. 89--176); C. Additive Number Theory, C1--C21 (pp. 159--208, PDF pp. 177--226); D. Diophantine Equations, D1--D29 (pp. 209--310, PDF pp. 227--328); E. Sequences of Integers, E1--E38 (pp. 311--364, PDF pp. 329--382); F. None of the Above, F1--F32 (pp. 365--404, PDF pp. 383--422); Index of Authors Cited (pp. 405--427, PDF pp. 423--445); General Index (pp. 429--437, PDF pp. 446--454). The sections the corpus cites, with their printed headings and starting pages:

• A2 Primes connected with factorials (p. 10, PDF p. 29); A5 Arithmetic progressions of primes (p. 25, PDF p. 44); A6 Consecutive primes in A.P. (p. 28, PDF p. 47); A8 Gaps between primes. Twin primes. (p. 31, PDF p. 50); A9 Patterns of primes (p. 40, PDF p. 59); A11 Increasing and decreasing gaps (p. 43, PDF p. 62); A13 Carmichael numbers (p. 50, PDF p. 69); A14 "Good" primes and the prime number graph (p. 54, PDF p. 73); A15 Congruent products of consecutive numbers (p. 54, PDF p. 73); A18 The Erdős--Selfridge classification of primes (p. 66, PDF p. 85); A19 Values of $n$ making $n-2^k$ prime. Odd numbers not of the form $\pm p^a\pm 2^b$. (p. 67, PDF p. 86).
• B2 Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers (p. 74, PDF p. 92); B3 Unitary perfect numbers (p. 84, PDF p. 102); B4 Amicable numbers (p. 86, PDF p. 104); B8 Unitary aliquot sequences (p. 97, PDF p. 115); B11 Solutions of $m\sigma(m)=n\sigma(n)$ (p. 101, PDF p. 119); B14 Some irrational series (p. 104, PDF p. 122); B15 Solutions of $\sigma(q)+\sigma(r)=\sigma(q+r)$ (p. 105, PDF p. 123); B16 Powerful numbers. Squarefree numbers. (p. 105, PDF p. 123); B18 Solutions of $d(n)=d(n+1)$ (p. 111, PDF p. 129); B19 $(m,n+1)$ and $(m+1,n)$ with same set of prime factors. The abc-conjecture. (p. 113, PDF p. 131); B21 $k\cdot 2^n+1$ composite for all $n$ (p. 119, PDF p. 137); B22 Factorial $n$ as the product of $n$ large factors (p. 122, PDF p. 140); B23 Equal products of factorials (p. 123, PDF p. 141); B24 The largest set with no member dividing two others (p. 124, PDF p. 142); B26 Densest set with no $l$ pairwise coprime (p. 125, PDF p. 143); B27 The number of prime factors of $n+k$ which don't divide $n+i$, $0\le i<k$ (p. 126, PDF p. 144); B29 Is $x$ determined by the prime divisors of $x+1$, $x+2$, ..., $x+k$? (p. 127, PDF p. 145); B30 A small set whose product is square (p. 128, PDF p. 146); B31 Binomial coefficients (p. 129, PDF p. 147); B32 Grimm's conjecture (p. 133, PDF p. 151); B33 Largest divisor of a binomial coefficient (p. 134, PDF p. 152); B35 Products of consecutive numbers with the same prime factors (p. 137, PDF p. 155); B36 Euler's totient function (p. 138, PDF p. 156); B37 Does $\phi(n)$ properly divide $n-1$? (p. 142, PDF p. 160); B38 Solutions of $\phi(m)=\sigma(n)$ (p. 144, PDF p. 162); B40 Gaps between totatives (p. 146, PDF p. 164); B41 Iterations of $\phi$ and $\sigma$ (p. 147, PDF p. 165); B42 Behavior of $\phi(\sigma(n))$ and $\sigma(\phi(n))$ (p. 150, PDF p. 168).
• C2 Sums of consecutive primes (p. 164, PDF p. 182); C4 Ulam numbers (p. 166, PDF p. 184); C5 Sums determining members of a set (p. 167, PDF p. 185); C8 Sets with distinct sums of subsets (p. 174, PDF p. 192); C9 Packing sums of pairs (p. 175, PDF p. 193); C10 Modular difference sets and error correcting codes (p. 181, PDF p. 199); C11 Three-subsets with distinct sums (p. 184, PDF p. 202); C15 Maximal zero-sum-free sets (p. 193, PDF p. 211); C16 Nonaveraging sets. Nondividing sets. (p. 198, PDF p. 216); C17 The minimum overlap problem (p. 199, PDF p. 217).
• D4 Waring's problem. Sums of $l$ $k$th Powers. (p. 229, PDF p. 247); D25 Equations involving factorial $n$ (p. 301, PDF p. 319).
• E1 A thin sequence with all numbers equal to a member plus a prime (p. 311, PDF p. 329); E2 Density of a sequence with l.c.m. of each pair less than $x$ (p. 312, PDF p. 330); E3 Density of integers with two comparable divisors (p. 313, PDF p. 331); E5 Sequence with members divisible by at least one of a given set (p. 315, PDF p. 333); E10 Theorem of van der Waerden. Szemerédi's theorem. Partitioning the integers into classes; at least one contains an A.P. (p. 317, PDF p. 335); E16 The $3x+1$ problem (p. 330, PDF p. 348); E28 $B_2$-sequences. Mian--Chowla sequences. (p. 350, PDF p. 368); E36 Klarner--Rado sequences (p. 361, PDF p. 379).
• F11 Distribution of residues of factorials (p. 381, PDF p. 399); F30 A polynomial whose sums of pairs of values are all distinct (p. 403, PDF p. 421).

Compiled scope

The book is compiled at statement depth for the passages the 95 citing problem pages consume, located as the rows below record. A row that says "named on the page" covers the section the problem page itself cites; a row that says "not named on the page" reports the section located here by a text-layer search of the whole book, as an identification made in this card and not by the problem page. Two pages cite the book for a question no passage of the text layer states (Problems 955 and 1065); their rows say so and name the nearest passages. The generated incoming-library block of each problem page derives from these rows; the problem pages themselves are not changed by this card.

Bears on. #1, not named on the page: C8 "Sets with distinct sums of subsets", p. 174 (PDF p. 192, text layer), Erdős's question for the maximum number $m$ of positive integers $a_1<\cdots<a_m\le 2^k$ with all subset sums distinct, the Erdős--Moser bounds $k+1\le m<k+\tfrac12\log k+2$ (Elkies lowering the 2 to $\tfrac12\log\pi<0.826$), the Conway--Guy sequence and its conjectured $m=k+2$, and the prize Erdős offered for a proof or disproof of $m=k+O(1)$. #3, not named on the page: A5 "Arithmetic progressions of primes", p. 25 (PDF p. 44, text layer), Erdős's conjecture, as the section reports it, that an infinite sequence of integers whose reciprocal sum diverges contains arbitrarily long arithmetic progressions, with his prize offer for a proof or disproof; E10, p. 318 (PDF p. 336) points back to it as the conjecture that would imply Szemerédi's theorem. #4, not named on the page: A8 "Gaps between primes. Twin primes.", p. 31 (PDF p. 50, text layer), Rankin's bound $d_n>c\ln n\ln\ln n\ln\ln\ln\ln n/(\ln\ln\ln n)^2$ for infinitely many $n$, Erdős's prize offer for proving or disproving that $c$ may be taken arbitrarily large, Rankin's $c=e^\gamma$, the Maier--Pomerance factor $1.31256$ and Pintz's $c=2e^\gamma$. #6, not named on the page: A11 "Increasing and decreasing gaps", p. 43 (PDF p. 62, text layer), Erdős and Turán's results that $d_n>d_{n+1}$ infinitely often and on a set of positive lower density, the section's statement that it is not known whether infinitely many runs of three consecutive values of $d_n$ decrease, or increase, and the prize offer for a proof that the alternating pattern cannot set in. #9, not named on the page: A19 "Values of $n$ making $n-2^k$ prime. Odd numbers not of the form $\pm p^a\pm2^b$.", p. 67 (PDF p. 86, text layer), Crocker's theorem that infinitely many odd integers are not of the form $2^k+2^l+p$ with $p$ prime, Erdős's suggestion that there may be $cx$ of them below $x$ together with the question whether even $>x^\epsilon$ can be proved, Erdős's questions whether, for each $r$, infinitely many odd integers are not the sum of a prime and $r$ or fewer powers of 2 and whether they have positive density, and Gallagher's theorem that for every $\epsilon>0$ and large enough $r$ the sums of a prime and $r$ powers of 2 have lower density greater than $1-\epsilon$. #11, not named on the page: A19, p. 67 (PDF p. 86, text layer), Erdős's further question whether some odd integer is not of the form $2^k+s$ with $s$ squarefree; Jud McCranie searched and found none below $1.4\cdot10^9$. #17, not named on the page: A8, p. 36 (PDF p. 55, text layer), the cluster primes, those $p$ for which every even number below $p-2$ is a difference of two primes both at most $p$, with 97 the smallest non-cluster prime and Blecksmith and Selfridge's question whether there are infinitely many; the same question is Erdős's second definition of a good prime in A14, p. 54 (PDF p. 73): $p$ is good if every even $2r\le p-3$ is a difference of two primes both at most $p$; 97 is the first prime that fails, and Selfridge and Blecksmith tabulated the good primes to $10^{37}$. #28, not named on the page: C9 "Packing sums of pairs", p. 177 (PDF p. 195, text layer), with $f(n)$ the number of solutions of $n=a_i+a_j$, the question whether some sequence has $\lim f(n)/\ln n=c$, the Erdős--Turán conjecture that $\limsup f(n)=\infty$ whenever $f(n)>0$ for all sufficiently large $n$, or whenever $a_k<ck^2$ for all $k$, and Erdős's prize offer for settling it. #30, not named on the page: C9, pp. 175--176 (PDF pp. 193--194, text layer), the maximum size $m$ of a Sidon sequence in $[1,n]$ with the upper bound due to Lindström (improving Erdős and Turán) and the lower bound due to Singer, the Erdős--Turán question whether $m=n^{1/2}+O(1)$ with Erdős's prize offer for settling it, and the constants $C>10.27$ (Zhang) and $C>13.71$ (Lindström) if $m<n^{1/2}+C$. #32, not named on the page: E1 "A thin sequence with all numbers equal to a member plus a prime", pp. 311--312 (PDF pp. 329--330; p. 312 on the page image), Erdős's prize offer for deciding whether a sequence thin enough that $A(x)<c\ln x$ can have every sufficiently large integer of the form $p+a_i$ with $p$ prime, with the analogs for squares (Moser, Erdős, Abbott, Balasubramanian and Soundararajan, Cilleruelo) and for powers of two (Ruzsa). #36, not named on the page: C17 "The minimum overlap problem", p. 199 (PDF p. 217, text layer), the definition of $M$, Erdős's $M>n/4$ with the improvements of Scherk, Świerczkowski and Moser, the Motzkin--Ralston--Selfridge examples with $M<2n/5$, which the section says went against Erdős's conjecture, the question whether $M\sim cn$ for some constant $c$, the table of $M(n)$ for $n\le15$ and Haugland's $\lim M(n)/n\le0.38200298812318988\ldots$. #39, not named on the page: C9, p. 176 (PDF p. 194, text layer), the infinite case: Erdős and Turán proved $\limsup a_k/k^2=\infty$ and gave a sequence with $\liminf a_k/k^2<\infty$; the dense infinite Sidon sequence of Ajtai, Komlós and Szemerédi with $a_k<ck^3/\ln k$ (printed $ck^3/\ln n$, a misprint), and the Erdős--Rényi sequence with $a_k<k^{2+\epsilon}$ and boundedly many representations. #41, not named on the page: C11 "Three-subsets with distinct sums", p. 184 (PDF p. 202, text layer), the $B_h$-sequences with the Bose--Chowla lower bound for $A_h(n)$ and, in the infinite case, Erdős's prize offer for a proof or disproof of $\liminf A_h(n)/n^{1/h}=0$, proved for $h=2$ by Erdős himself and for $h=4$ by Nash, with Jia treating $h=6$, Chen settling every even $h$ and odd $h$ reported open; E28, p. 351 (PDF p. 369) repeats the case $h=3$ as Erdős's prize offer for a proof or disproof of his old conjecture $\lim a_n/n^3=\infty$. #44, not named on the page: C9, p. 177 (PDF p. 195, text layer), after the perfect-difference-set question, Erdős's admission that he "could not even decide" whether a finite Sidon sequence can be prolonged to $a_1<a_2<\cdots<a_k<a_{k+1}<\cdots<a_n$ with $a_n<(1+o(1))n^2$, that is, made asymptotically as dense as possible. #45, named on the page (B2): B2 "Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers", p. 80 (PDF p. 98, page image), Erdős's $n_k$, the smallest integer such that every partition of the proper divisors of $n_k$ into $k$ classes leaves $n_k$ a sum of distinct divisors from one class; $n_1=6$, and the section says Erdős could not even prove that $n_2$ exists; the divisor-sum form of the page's unit-fraction question (a sum of distinct divisors of $n_k$ equal to $n_k$ is a sum of unit fractions with denominators dividing $n_k$ equal to 1). #48, not named on the page: B38 "Solutions of $\phi(m)=\sigma(n)$", p. 144 (PDF p. 162, text layer), the question whether infinitely many pairs $m$, $n$ have $\phi(m)=\sigma(n)$, answered by infinitely many twin primes or infinitely many Mersenne primes, with sporadic solutions such as $\phi(780)=192=\sigma(105)$. #51, not named on the page: B36 "Euler's totient function", p. 139 (PDF p. 157, text layer), Erdős's question, as the section reports it, whether for every $\epsilon$ there is an $n$ with $\phi(n)=m$, $m<\epsilon n$, and $\phi(t)\ne m$ for every $t<n$, with the suggestion that there may be many such $n$. #56, not named on the page: B26 "Densest set with no $l$ pairwise coprime", p. 125 (PDF p. 143, page image), Erdős's question for the maximum $k$ such that some integers $1\le a_1<a_2<\cdots<a_k\le n$ have no $l$ among them pairwise relatively prime, his conjecture that the maximum is the number of integers $\le n$ divisible by one of the first $l-1$ primes, and the section's report that he called $l=2$ easy and $l=3$ not difficult and offered a prize for a general solution. #131, named on the page (C16): C16 "Nonaveraging sets. Nondividing sets.", p. 198 (PDF p. 216, page image), Erdős's original question for the maximum number $k(x)$ of integers in $[0,x]$ none of which divides the sum of any of the others; a nondividing set is nonaveraging, so $k(x)\le f(x)$, and Straus showed $k(x)\ge\max\{f(x/f(x)),f(\sqrt x)\}$. #141, not named on the page: A6 "Consecutive primes in A.P.", p. 28 (PDF p. 47, text layer), the conjecture, reported there, that there are arbitrarily long arithmetic progressions of consecutive primes, with the examples 251, 257, 263, 269, the five- and six-term progressions of Jones, Lal and Blundon and of Lander and Parkin, and the seven consecutive primes with common difference 210 found by Dubner and Nelson in 1995. #144, not named on the page: E3 "Density of integers with two comparable divisors", p. 313 (PDF p. 331, text layer), the question whether the integers with two divisors $d_1<d_2<2d_1$ have density one; Erdős had shown that the density exists, and the section records that Maier and Tenenbaum answered yes since the first edition. #175, not named on the page: B33 "Largest divisor of a binomial coefficient", p. 135 (PDF p. 153, text layer), Erdős's conjecture that $\binom{2n}n$ is never squarefree for $n>4$, Sárközy's proof for $n$ sufficiently large, Sander's result near the center of Pascal's triangle, and Granville and Ramaré's completion of Sárközy's proof, showing that $k>300000$ is large enough and checking $2\le k\le300000$ by computer. #186, named on the page (C16): C16, p. 198 (PDF p. 216, page image), the nonaveraging sets of Erdős and Straus with $f(x)$ their maximum size, the bounds $\tfrac14\log x+O(1)<\log f(x)<\tfrac12(\log x+\log\ln x)+O(1)$ (logarithms to base 2), the conjecture $f(x)=\exp(c\sqrt{\ln x})=o(x^\epsilon)$, and Abbott's $l(n)>n^{1/13-\epsilon}$ for the largest nonaveraging subset every set of $n$ integers contains. #219, not named on the page: A5, p. 25 (PDF p. 44, text layer), the conjecture that the length $n$ of an arithmetic progression of primes can be arbitrarily large, which would follow from an improvement of Szemerédi's theorem (see E10), and the dated note "STOP PRESS (04-04-09)" reporting that Green and Tao had obtained an improvement of this kind and that, in Guy's judgment, they had almost certainly proved the conjecture. #220, not named on the page: B40 "Gaps between totatives", p. 146 (PDF p. 164, text layer), Erdős's conjecture $\sum(a_{i+1}-a_i)^2<cn^2/\phi(n)$ with his prize offer for a proof, Hooley's $\sum(a_{i+1}-a_i)^2\ll n(\ln\ln n)^2$, Vaughan's proof of the conjecture on average, and the prize won by Montgomery and Vaughan. #233, not named on the page: A8, pp. 33--34 (PDF pp. 52--53, text layer), Cramér's bound $\sum_{n<x}d_n^2<cx(\ln x)^4$ under the Riemann hypothesis, and Erdős's conjecture that the right-hand side should be $cx(\ln x)^2$, which the section says he thought hopeless to prove. #241, not named on the page: C11, p. 184 (PDF p. 202, text layer), the Bose--Chowla lower bound for $A_h(n)$, the upper bounds of Jia (even $h$), Chen and Graham (odd $h$) and Graham's further small improvement for $h=3$, $A_3(n)\le(4-\tfrac1{228})^{1/3}n^{1/3}(1+o(1))$ (a display read on the page image, printed after the next sentence), and Helm's result that no sequence with $A(n)\sim\alpha n^{1/3}$ terms can be a $B_3$-sequence, with no value of $\alpha$ printed. #252, named on the page (B14): B14 "Some irrational series", p. 104 (PDF p. 122, page image), the question whether $\sum_{n=1}^\infty\sigma_k(n)/n!$ is irrational, known for $k=1$ and 2, followed by Erdős's irrationality of $\sum1/(2^n-1)=\sum d(n)/2^n$ and Borwein's series. #322, not named on the page: D4 "Waring's problem. Sums of $l$ $k$th Powers.", p. 229 (PDF p. 247, text layer), $r_{k,l}(n)$ the number of representations as a sum of $l$ $k$th powers, Hardy and Littlewood's Hypothesis K, Mahler's disproof for $k=3$ with $r_{3,3}>c_1n^{1/12}$ for infinitely many $n$, Erdős's view that $r_{3,3}<c_2n^{1/12}$ may hold for all $n$ though nothing is known, and the Chowla--Erdős bound $r_{k,k}>\exp(c_k\ln n/\ln\ln n)$ for infinitely many $n$. #324, not named on the page: F30 "A polynomial whose sums of pairs of values are all distinct", p. 403 (PDF p. 421, text layer), Erdős's unsolved problem of finding a polynomial $P(x)$ with all sums $P(a)+P(b)$, $0\le a<b$, distinct, $x^5$ being the section's likely answer, with Ruzsa's almost polynomial Sidon set $\{\lfloor n^5+\xi n^4\rfloor\}$. #342, not named on the page: C4 "Ulam numbers", p. 166 (PDF p. 184, text layer), the U-numbers and five questions on them, introduced as some of those Recamán asked, among them whether they have positive density, a question marked as Ulam's, and whether there are infinitely many pairs of consecutive U-numbers, with Muller's 20000 terms, more than 60% of which differ from another term by exactly 2. #358, named on the page (C2): C2 "Sums of consecutive primes", p. 164 (PDF p. 182, text layer and page image), Erdős's question whether some infinite integer sequence $1<a_1<a_2<\cdots$ has its count $f(n)$ of solutions of $a_i+a_{i+1}+\cdots+a_k=n$ tending to infinity with $n$; his note that with $k>i$ required it is not even known whether $f(n)>0$ for all but finitely many $n$; and the example $a_i=i$, where $f(n)$ is the number of odd divisors of $n$; the question follows Moser's questions on $f(n)$ for sums of consecutive primes, the section's only reference is Moser 1963, and none is given for the Erdős question. #365, not named on the page: B16 "Powerful numbers. Squarefree numbers.", pp. 105--106 (PDF pp. 123--124, text layer), Golomb's infinitely many pairs of consecutive powerful numbers, Erdős's $k$-full numbers $u_i^{(k)}$, his question whether $u_{i+1}^{(2)}-u_i^{(2)}=1$ has infinitely many solutions not coming from Pell equations $x^2-dy^2=\pm1$, and whether some constant $c$ bounds the number of solutions with $u_i<x$ by $(\ln x)^c$. #366, not named on the page: B16, p. 106 (PDF p. 124, text layer), the question whether $u_{i+1}^{(3)}-u_i^{(3)}=1$ has no solutions, that is, whether no two consecutive integers are both 3-full, with the companion question on simultaneous solutions and Mąkowski's answers to some of Erdős's other questions. #373, not named on the page: B23 "Equal products of factorials", p. 123 (PDF p. 141, text layer), the equation $n!=a_1!a_2!\cdots a_r!$ with $r\ge2$ and $a_1\ge a_2\ge\cdots\ge a_r\ge2$, the trivial family, Hickerson's $9!=7!3!3!2!$, $10!=7!6!=7!5!3!$ and $16!=14!5!2!$, the searches to 18160 and $10^6$, and Erdős's observation that if $P(n)$, the largest prime factor of $n$, were known to satisfy $P(n(n+1))/\ln n\to\infty$, only finitely many nontrivial examples could exist. #375, not named on the page: B32 "Grimm's conjecture", p. 133 (PDF p. 151, text layer), Grimm's conjecture that if $n+1$, $n+2$, ..., $n+k$ are all composite there are distinct primes $p_{i_j}$ with $p_{i_j}\mid(n+j)$ for $1\le j\le k$, with two examples, and the theorem of Ramachandra, Shorey and Tijdeman that under the hypothesis of Schinzel mentioned in A2 the conjecture has only finitely many exceptions. #376, not named on the page: B33, p. 135 (PDF p. 153, text layer), Ron Graham's prize offer for deciding whether $(\binom{2n}n,105)=1$ infinitely often, Kummer's criterion restricting such $n$ to the digits 0, 1 in base 3, 0, 1, 2 in base 5 and 0, 1, 2, 3 in base 7, the 14 values of $n<7^{10}$ found by Gupta and Khare, and the Erdős--Graham--Ruzsa--Straus result for two primes. #377, not named on the page: B33, p. 135 (PDF p. 153, text layer), with $f(n)$ the sum of the reciprocals of the primes $<n$ that do not divide $\binom{2n}n$, the Erdős--Graham--Ruzsa--Straus conjecture that $f(n)<c$ for all $n$ with an absolute constant $c$. #384, named on the page (B31 and B33): B31 "Binomial coefficients", pp. 129--130 (PDF pp. 147--148; p. 129 on the page image), Ecklund, Eggleton, Erdős and Selfridge's factorization $\binom nk=UV$ with the prime factors of $U$ at most $k$ and those of $V$ greater than $k$, finitely many cases $n\ge2k$ with $U>V$; B33, p. 134 (PDF p. 152, page image), Ecklund's theorem that $\binom nk$ with $n\ge2k>2$ has a prime divisor $p\le n/2$ except for $\binom73$, Faulkner's theorem with the exceptions $\binom92$ and $\binom{10}3$, and Selfridge's conjecture that for $n\ge k^2-1$, apart from $\binom{62}6$, $\binom nk$ has a prime divisor $\le n/k$. #387, not named on the page: B33, p. 134 (PDF p. 152, page image), the question what can be said about the largest divisor below $n$ of $\binom nk=n!/k!(n-k)!$; Erdős's remark that it is easily at least $n/k$, and his conjecture that for any $c<1$ and $n$ sufficiently large there is one between $cn$ and $n$. #391, not named on the page: B22 "Factorial $n$ as the product of $n$ large factors", p. 122 (PDF p. 140, text layer), the Straus--Erdős--Selfridge problem of writing $n!$ as a product of $n$ factors with the least factor $l$ as large as possible, the example $n=56$, $l=15$, Selfridge's two conjectures ($l\ge\lfloor2n/7\rfloor$ except for $n=56$; $l\ge n/3$ for $n\ge300000$), the report that Straus was said to have shown $l>n/(e+\epsilon)$ for $n>n_0(\epsilon)$ but that no proof turned up in his Nachlass, and Erdős's questions on the gaps and constant stretches in the values of $l$. #399, not named on the page: D25 "Equations involving factorial $n$", p. 301 (PDF p. 319, text layer), the Erdős--Obláth treatment of $n!=x^p\pm y^p$ with $x\perp y$ and $p>2$, with the $p=2$ cases referred to D2 and to the factorization of $n!$ into two even factors. #406, not named on the page: B33, p. 135 (PDF p. 153, text layer), Erdős's conjecture that for $k>8$, $2^k$ is not a sum of distinct powers of 3 (the section notes $2^8=3^5+3^2+3+1$), the base-3 form of the page's question, stated there for its consequence, a display read on the page image, that $3\mid\binom{2^{k+1}}{2^k}$ for $k\ge9$. #408, not named on the page: B41 "Iterations of $\phi$ and $\sigma$", pp. 147--148 (PDF pp. 165--166, text layer), the class $k(n)$, the least $k$ with $\phi_k(n)=1$, Pillai's bounds between $\ln n/\ln3$ and $\ln n/\ln2$, the density of $k(n)/\ln n$ in $[1/\ln3,1/\ln2]$, the question of the average and normal behavior of $k(n)$, the Erdős--Granville--Pomerance--Spiro conjecture that $k(n)$ has normal order $\alpha\ln n$ for some constant $\alpha$, which they prove under the Elliott--Halberstam conjecture, and their normal order $he^\gamma\ln\ln\ln n$ for $\phi_h(n)/\phi_{h+1}(n)$. #409, not named on the page: B41, p. 148 (PDF p. 166, text layer), Finucane's iteration of $\phi(n)+1$ and his questions: in how many steps a prime is reached; for a prime $p$, how the $n$ whose sequences end in $p$ are distributed; whether 5, 8, 10, 12 are the only numbers leading to 5, and 7, 9, 14, 15, 16, 18, 20, 24, 30 the only ones leading to 7. #410, not named on the page: B41, pp. 148--149 (PDF pp. 166--167, text layer), among the six statements on the iterates $\sigma^k(n)$ that Erdős, Granville, Pomerance and Spiro could neither prove nor disprove, whether $(\sigma^k(n))^{1/k}\to\infty$ as $k\to\infty$ for every $n>1$, and Erdős's remark that for the iterations of $\sigma(n)-1$ and of the two averages he cannot show growth slower than exponential. #413, not named on the page: B8 "Unitary aliquot sequences", p. 98 (PDF p. 116, text layer), the Erdős--Selfridge definition of a barrier for a number-theoretic function $f(m)$, an $n$ with $m+f(m)\le n$ for all $m<n$; Euler's $\phi$ (see B36) and $\sigma(m)$ grow too fast to have barriers; the questions whether $\omega(m)$, with the barriers 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 20, 24, 26, 28, 30, ..., and $\Omega(m)$ have infinitely many barriers, with Selfridge's 99840 and Mąkowski's remarks on $d(m)$. #416, not named on the page: B36, p. 139 (PDF p. 157, text layer), Erdős and Hall's $\Phi(y)=ye^{f(y)}/\ln y$ for the number of $n\le y$ with $\phi(x)=n$ solvable, $f(y)$ between $c(\ln\ln\ln y)^2$ and $c(\ln y)^{1/2}$, Maier and Pomerance's proof that the lower bound is correct with $c\approx0.8178$, and Erdős's conjecture $\Phi(cy)/\Phi(y)\to c$, which he suggested may, if true, be as close as one can come to an asymptotic formula for $\Phi(y)$. #418, not named on the page: B36, p. 139 (PDF p. 157, text layer), Browkin and Schinzel's proof of the Sierpiński--Erdős conjecture that there are infinitely many noncototients, the $n$ for which $x-\phi(x)=n$ has no solution, by showing that no number $2^k\cdot509203$, $k=1,2,\ldots$, is of the form $x-\phi(x)$. #441, named on the page (B26 and E2): B26, p. 125 (PDF p. 143, page image), the dual question for the largest subset of $[1,n]$ with every pairwise least common multiple at most $n$; with $g(n)$ its size, Erdős's bounds $\frac3{2\sqrt2}n^{1/2}-2<g(n)\le2n^{1/2}$, the lower one from the integers up to $(n/2)^{1/2}$ plus the even integers between $(n/2)^{1/2}$ and $(2n)^{1/2}$, and Choi's upper bound $1.638n^{1/2}$; E2 "Density of a sequence with l.c.m. of each pair less than $x$", p. 312 (PDF p. 330, page image), the bounds $(9x/8)^{1/2}\le\max A(x)\le(4x)^{1/2}$, with the same construction and Erdős's further questions on $B(x)$ and $C(x)$. #453, not named on the page: A14 '"Good" primes and the prime number graph', p. 54 (PDF p. 73, text layer), the Erdős--Straus good primes, those $p_n$ with $p_n^2>p_{n-i}p_{n+i}$ for all $1\le i\le n-1$ (for example 5, 11, 17 and 29), Pomerance's proof by the prime number graph (see A5) that there are infinitely many, and his further questions on their density and on the sums and products of neighboring primes. #470, not named on the page: B2, p. 77 (PDF p. 95, text layer), Benkoski's weird numbers, abundant but not pseudoperfect, the 24 primitive weird numbers below a million, Benkoski and Erdős's positive density, and the open questions whether infinitely many primitive abundant numbers are weird, whether every odd abundant number is pseudoperfect (not weird), and whether $\sigma(n)/n$ can be arbitrarily large for weird $n$. #476, named on the page (C15): C15 "Maximal zero-sum-free sets", p. 194 (PDF p. 212, text layer; the section opens on p. 193, PDF p. 211, page image), the Erdős--Heilbronn theorem that $k\ge3(6p)^{1/2}$ distinct residues $a_1,a_2,\ldots,a_k$ modulo a prime $p$ represent every residue mod $p$ as $\sum_{i=1}^k\epsilon_ia_i$ with $\epsilon_i\in\{0,1\}$; their conjecture that $k>2\sqrt p$ suffices and is best possible, proved by Olson (printed "Olsen"); their further conjecture that the number $s$ of distinct residues $a_i+a_j$, $1\le i<j\le k$, is at least $\min\{p,2k-3\}$, with partial results of Mansfield, of Rødseth and of Freiman, Low and Pitman; and the complete proof by Dias da Silva and Hamidoune, who showed that for $A\subseteq\mathbb Z/p\mathbb Z$ with $|A|=k$ the set $A^h$ of sums of $h$ distinct elements of $A$ has $|A^h|\ge\min\{p,hk-h^2+1\}$, with Nathanson's simplification and the Nathanson--Ruzsa bound for two sets. #478, not named on the page: F11 "Distribution of residues of factorials", p. 381 (PDF p. 399, text layer), the question how $1!$, $2!$, $3!$, ..., $(p-1)!$, $p!$ are distributed modulo $p$, about $p/e$ of the residue classes being missed, the table of missing residues for $p\le37$, and Rokowska and Schinzel's answer to an Erdős question on primes with $2!,\ldots,(p-1)!$ all distinct mod $p$. #488, named on the page (E5): E5 "Sequence with members divisible by at least one of a given set", p. 315 (PDF p. 333, page image), with $D(x)$ the number of integers $\le x$ divisible by at least one term of a finite sequence $a_1<a_2<\cdots<a_k\le n$, the question whether $D(x)/x<2D(n)/n$ for all $x>n$; the constant 2 cannot be lowered, as $n=2a_1-1$, $x=2a_1<a_2$ shows; in the other direction, for each $\epsilon>0$ some sequence fails $D(x)/x>\epsilon D(n)/n$; with Besicovitch 1934 and Erdős 1935 as its references. #494, not named on the page: C5 "Sums determining members of a set", pp. 167--168 (PDF pp. 185--186, text layer), Leo Moser's question, largely settled by Selfridge, Straus and others, of how far the pairwise sums of a set determine the set: they do when the cardinality is not a power of two; the power-of-two ambiguities with the three eight-element examples; Boman and Linusson's settlement of the problem for sums of triples, the exceptions being exactly 3, 6, 27, 486; and Ewell's settlement for sums of four distinct elements. #530, not named on the page: C9, p. 177 (PDF p. 195, text layer), the question whether every sequence of integers $a_1<a_2<\cdots<a_n$ contains a Sidon subsequence $a_{i_1},\ldots,a_{i_m}$ with $m=(1+o(1))n^{1/2}$, which Komlós, Sulyok and Szemerédi (see E11) proved with $m>cn^{1/2}$, and, pp. 177--178, Abbott's $g(m)>cm^{1/2}$, for any constant $c<\tfrac2{25}$ and all sufficiently large $m$, for the largest Sidon subset every set of $m$ integers contains. #540, named on the page (C15): C15, pp. 193--194 (PDF pp. 211--212; p. 193 on the page image), the Erdős--Heilbronn question for the largest number $k=k(m)$ of distinct residue classes modulo $m$ with no subset summing to zero, the example $k(20)=6$, the bound $k\ge\lfloor(-1+\sqrt{8m+9})/2\rfloor$ for $m>5$ with equality for $5<m\le24$, Selfridge's construction for $m=2(l^2+l+1)$ giving $k\ge2l+1=\sqrt{2m-3}$ and his conjecture for even $m$, his further conjecture that $k(p)=k$ for primes $\tfrac12k(k+1)<p<\tfrac12(k+1)(k+2)$, with the case $k(43)=8$ confirmed by Lam, and the question whether $k=\lfloor(-1+\sqrt{8m+9})/2\rfloor$ for infinitely many $m$. #677, named on the page (B35): B35 "Products of consecutive numbers with the same prime factors", p. 138 (PDF p. 156, page image), with $L(n;k)$ the l.c.m. of $n+1$, $n+2$, ..., $n+k$, Erdős's conjecture that for $l>1$ and $n\ge m+k$ the equation $L(m;k)=L(n;l)$ has only finitely many solutions, with the examples $L(4;3)=L(13;2)$ and $L(3;4)=L(19;2)$, followed by the questions on $L(n;k)>L(n-k;k)$ and the largest $k=k(n)$ reversing it, with $k(n)=o(n)$ easy and $k(n)<n^{1/2+\epsilon}$ expected; no proof is given for the finiteness claim, and the section's only reference is Erdős's 1980 Monthly note. #699, not named on the page: B31, p. 131 (PDF p. 149, text layer), the question whether $\binom nr$ and $\binom ns$, $0<r<s\le n/2$, are ever coprime, answered no by an identity lost in the text layer, and the Erdős--Szekeres question whether the greatest prime factor of the g.c.d. always exceeds $r$, the only counterexample with $r>3$ they noticed being a display lost in the text layer. #707, not named on the page: C9, pp. 176--177 (PDF pp. 194--195, text layer), Erdős's question whether a Sidon sequence $a_1<a_2<\cdots<a_k$ can be extended to a perfect difference set (see C10), one whose differences $a_u-a_v$, $1\le u,v\le p+1$, $u\ne v$, represent every nonzero residue mod $p^2+p+1$ exactly once; C10 "Modular difference sets and error correcting codes", p. 181 (PDF p. 199), asks in general whether a finite sequence with no repeated differences can always be extended to a perfect difference set, after Singer's existence theorem for prime powers $k$ and the conjecture that no perfect difference set exists otherwise. #825, not named on the page: B2, p. 77 (PDF p. 95, text layer), the last of the weird-number questions, whether $\sigma(n)/n$ can be arbitrarily large for weird $n$; Benkoski and Erdős expect the answer no, and Erdős put up prizes for this question and for the one before it (whether every odd abundant number is pseudoperfect); weird means abundant and not the sum of a set of its divisors, so the page's constant $C$ is the conjectured bound on $\sigma(n)/n$. #828, not named on the page: B37 "Does $\phi(n)$ properly divide $n-1$?", pp. 142--143 (PDF pp. 160--161, text layer), Lehmer's eight solutions of $\phi(n)\mid n+1$ and Ron Graham's conjecture that for every $k$ there are infinitely many $n$ with $\phi(n)\mid(n-k)$, which he observed holds for $k=0$, for $k=2^a$ ($a\ge0$) and for $k=2^a3^b$ ($a,b>0$); Pomerance, in the Acta Arith. paper cited at B2, treated Graham's problem. #830, not named on the page: B4 "Amicable numbers", pp. 86--87 (PDF pp. 104--105, text layer), infinitely many amicable pairs believed but not known to exist, Erdős's conjecture that the number $A(x)$ of pairs with $m<n<x$ is at least $x^{1-\epsilon}$, his improvement of a result of Kanold to $A(x)=o(x)$, Pomerance's $A(x)\le x\exp\{-(\ln x)^{1/3}\}$ (so the reciprocal sum converges) and $A(x)\ll x\exp\{-c(\ln x\ln\ln x)^{1/3}\}$, and te Riele's 1427 pairs with lesser member below $10^{10}$. #841, not named on the page: B30 "A small set whose product is square", pp. 128--129 (PDF pp. 146--147, text layer; p. 129 on the page image), the Erdős--Graham--Selfridge problem of the least $t_n$ such that $n+1$, $n+2$, ..., $n+t_n$ contain a subset whose product with $n$ is a square, the section's statement that the Thue--Siegel theorem gives $t_n\to\infty$ faster than a power of $\ln n$, with Granville's and Silverman's comments on that sentence, and Selfridge's bound $t_n\le\max(P(n),3\sqrt n)$ with $P(n)$ the largest prime factor of $n$. #850, not named on the page: B29 "Is $x$ determined by the prime divisors of $x+1$, $x+2$, ..., $x+k$?", p. 127 (PDF p. 145, text layer), Alan R. Woods's question whether some positive integer $k$ makes every $x$ determined by the sets of prime divisors of $x+1$, $x+2$, ..., $x+k$, perhaps $k=3$, with the four ambiguous cases for $k=2$ involving only primes below 23 and the infinite family $(2^n-2,2^n-1)$, $(2^n(2^n-2),(2^n-1)^2)$; B19, p. 113 (PDF p. 131) has Erdős's two-term question for $m,n$ and $m+1,n+1$ with Mąkowski's pair $m=3\cdot5^2$, $n=3^5\cdot5$. #855, not named on the page: A9 "Patterns of primes", p. 40 (PDF p. 59, text layer), the prime-pattern conjecture, which the section says is incompatible with the Hardy--Littlewood conjecture $\pi(x+y)\le\pi(x)+\pi(y)$ for all integers $x,y\ge2$; Guy sets extra query marks round the latter because, in his words, "it is very likely to be false"; with the alternative $\pi(x+y)\le\pi(x)+2\pi(y/2)$ and the Montgomery--Vaughan bound $\pi(x+y)-\pi(x)\le2y/\ln y$. #860, not named on the page: B32, p. 133 (PDF p. 151, text layer), the Erdős--Selfridge function $f(n)$, the least length for which every interval $[m+1,m+f(n)]$ contains distinct integers $a_1,a_2,\ldots,a_{\pi(n)}$ with $p_i\mid a_i$, $p_i$ the $i$th prime; they and Pomerance show $(3-\epsilon)n\le f(n)\ll n^{3/2}(\ln n)^{-1/2}$ for large $n$. #861, not named on the page: C9, p. 176 (PDF p. 194, text layer), the Cameron--Erdős problem of estimating $F(n)$, the count of Sidon sequences contained in $[1,n]$; with $m$ the maximum size of a Sidon sequence in $[1,n]$ as above, the section records $\limsup F(n)/2^m=\infty$ as known and $F(n)/2^m\to\infty$ as open; Cameron and Erdős expect $F(n)<n^{\epsilon\sqrt n}$; with partial results of Alon and of Calkin and Thomson and the modular bounds of Lev and Schoen. #889, not named on the page: B27 "The number of prime factors of $n+k$ which don't divide $n+i$, $0\le i<k$", p. 126 (PDF p. 144, text layer), the Erdős--Selfridge $v(n;k)$, the number of prime factors of $n+k$ dividing none of $n+i$, $0\le i<k$, and $v_0(n)$, the maximum of $v(n;k)$ over all $k\ge0$; the question whether $v_0(n)\to\infty$ with $n$; their result that $v_0(n)>1$ for all $n$ except 1, 2, 3, 4, 7, 8 and 16; with the variants $v_l(n)$ and $V(n;k)$. #931, not named on the page: B35, p. 138 (PDF p. 156, page image), Erdős's question whether $(m+1)(m+2)\cdots(m+k)$ and $(n+1)(n+2)\cdots(n+l)$ with $k\ge l\ge3$ can have the same prime factors infinitely often, with the examples $2\cdot3\cdots10$ against $14\cdot15\cdot16$ and against $48\cdot49\cdot50$, and $2\cdot3\cdots12$ against $98\cdot99\cdot100$, and his conjecture that for $k=l\ge3$ this happens only finitely many times, with Erdős's 1980 Monthly note as the reference. #945, not named on the page: B18 "Solutions of $d(n)=d(n+1)$", p. 112 (PDF p. 130, text layer), the Erdős--Mirsky question for the largest $k$ such that $d(n)$, $d(n+1)$, ..., $d(n+k)$ are all distinct; only trivial bounds are known, and the section guesses $k=(\ln n)^c$. #946, not named on the page: B18, pp. 111--112 (PDF pp. 129--130, text layer), Claudia Spiro's proof that $d(n)=d(n+5040)$ has infinitely many solutions, Heath-Brown's use of her ideas to show that $d(n)=d(n+1)$ has infinitely many solutions, and Pinner's extension to $d(n)=d(n+a)$ for every integer $a$, with the counts $\ll x/(\ln\ln x)^{1/2}$ (Erdős, Pomerance and Sárközy) and $\gg x/(\ln\ln x)^3$ (Hildebrand). #955, not named on the page: no passage stating that a density-zero set has a density-zero preimage under $s(n)=\sigma(n)-n$ was found in the text layer (searched for density, aliquot, $s(n)$ and Pomerance in part B); the nearest passage is B10 "Untouchable numbers", p. 100 (PDF p. 118), Erdős's proof that $s(x)=n$ has no solution for infinitely many $n$, and in fact that the untouchable numbers have positive lower density. #1052, not named on the page: B3 "Unitary perfect numbers", pp. 84--85 (PDF pp. 102--103, text layer), no odd unitary perfect number exists, Subbarao conjectures that there are only finitely many even ones, Subbarao, Carlitz and Erdős have each put up prizes for a resolution and Subbarao offers a reward per newly found example, with the four small examples, Wall's fifth and the exhausted ranges $a\le10$ and $r\le6$ for $n=2^am$. #1053, not named on the page: B2, p. 78 (PDF p. 96, text layer), the multiperfect numbers $\sigma(n)=kn$, the counts known at the end of 2002 (5000 for $3\le k\le11$, besides 1 for $k=1$ and 39 for $k=2$; the book gives the overall total as "8!", though these add to $5040=7!$), the question whether $k$ can be arbitrarily large, Erdős's conjecture $k=o(\ln\ln n)$, and the suggestion that there may be only finitely many $k$-perfect numbers for each $k\ge3$. #1054, not named on the page: B2, p. 80 (PDF p. 98, page image), Erdős's $f(n)$, the smallest integer whose divisors $1=d_1<d_2<\ldots<d_l=f(n)$ satisfy $n=\sum_{i=1}^kd_i$ for some $k$; the questions whether $f(n)=o(n)$, or whether that holds only for almost all $n$ with $\limsup f(n)/n=\infty$; with a table of $f(n)$ for $n\le28$. #1055, not named on the page: A18 "The Erdős--Selfridge classification of primes", p. 66 (PDF p. 85, text layer), the classification: $p$ is in class 1 if $p+1$ has no prime divisor other than 2 and 3, and in class $r$ if every prime factor of $p+1$ lies in a class $\le r-1$ with at least one in class $r-1$; the tables of classes 1--8; the easy bound $o(n^\epsilon)$, for every $\epsilon>0$ and every $r$, on the number of class-$r$ primes up to $n$; the problem of proving every class infinite; and, with $p_1^{(r)}$ the least prime of class $r$ ($p_1^{(1)}=2$, $p_1^{(2)}=13$, $p_1^{(3)}=37$, $p_1^{(4)}=73$, $p_1^{(5)}=1021$), Erdős's expectation that $(p_1^{(r)})^{1/r}\to\infty$ against Selfridge's that it is quite likely bounded. #1056, not named on the page: A15 "Congruent products of consecutive numbers", p. 54 (PDF p. 73, text layer), Erdős's observation, in a letter dated 79-10-31, that $3\cdot4\equiv5\cdot6\cdot7\equiv1\bmod11$, his question for the least prime $p$ admitting integers $a$, $k_1$, $k_2$, $k_3$ with $\prod_{i=1}^{k_1}(a+i)\equiv\prod_{i=1}^{k_2}(a+k_1+i)\equiv\prod_{i=1}^{k_3}(a+k_1+k_2+i)\equiv1\bmod p$, and his suggestion that such primes exist for any number of congruent products, with the examples of Mąkowski and Narkiewicz and the Noll--Simmons table. #1057, not named on the page: A13 "Carmichael numbers", p. 50 (PDF p. 69, text layer), Alford, Granville and Pomerance's infinitely many Carmichael numbers, more than $x^\beta$ below $x$ with $\beta>0.290306>2/7$, Erdős's conjecture that $(\ln C(x))/\ln x\to1$ as $x\to\infty$ and his improvement of a result of Knödel to $C(x)<x\exp\{-c\ln x\ln\ln\ln x/\ln\ln x\}$, Pomerance, Selfridge and Wagstaff's $c=1-\epsilon$ and their heuristic for the reverse inequality with $c=2+\epsilon$, and Pinch's counts to $10^{16}$. #1058, not named on the page: A2 "Primes connected with factorials", p. 11 (PDF p. 30, text layer), the Erdős--Stewart conjecture that $1!+1=2$, $2!+1=3$, $3!+1=7$, $4!+1=5^2$, $5!+1=11^2$ are the only cases of $n!+1=p_k^ap_{k+1}^b$ with $p_{k-1}\le n<p_k$, and the announcement by Flammenkamp and Luca on 98-12-16 that they had proved it. #1059, not named on the page: A2, p. 11 (PDF p. 30, text layer), Erdős's question whether infinitely many primes $P$ have $P-k!$ composite for every $k$ with $1\le k!<P$, for example $P=101$ and $P=211$, and his suggestion that the easier target is infinitely many integers $n$ with $i!<n\le(i+1)!$, all prime factors greater than $i$, and every $n-k!$ ($1\le k\le i$) composite. #1060, not named on the page: B11 "Solutions of $m\sigma(m)=n\sigma(n)$", pp. 101--102 (PDF pp. 119--120, text layer), Moser's observation that $n\sigma(n)$ does not determine $n$, Erdős's results that $n\sigma(n)$ is distinct for squarefree $n$ and that the number of solutions with $m<n<x$ is $cx+o(x)$, and his belief that $x\sigma(x)=n$ has fewer than $n^{\epsilon/\ln\ln n}$ solutions for every $\epsilon>0$, perhaps fewer than $(\ln n)^c$. #1061, not named on the page: B15 "Solutions of $\sigma(q)+\sigma(r)=\sigma(q+r)$", p. 105 (PDF p. 123, text layer), Rumney's question and the families of primitive solutions, then Erdős's questions: how many solutions, not necessarily primitive, have $q+r<x$, whether $cx+o(x)$ or of higher order; and, with $s_1<s_2<\cdots$ the numbers for which $\sigma(s_i)=\sigma(q)+\sigma(s_i-q)$ has a solution with $q<s_i$, the density of the sequence $\{s_i\}$. #1062, not named on the page: B24 "The largest set with no member dividing two others", p. 124 (PDF p. 142, text layer), the largest size $f(n)$ of a subset of $[1,n]$ in which no member divides two others, with Erdős's question how large it can be, the $\lceil2n/3\rceil$ example, Kleitman's $f(29)=21$, Lebensold's $0.6725n\le f(n)\le0.6736n$ for large $n$, and Erdős's question whether $\lim f(n)/n$ is irrational. #1063, not named on the page: B31, p. 130 (PDF p. 148, text layer), the Erdős--Selfridge observation that for $n\ge2k\ge4$ some $i$ with $0\le i\le k-1$ has $n-i\nmid\binom nk$, and their question for the least $n_k$ with only one such $i$; $n_2=4$, $n_3=6$, $n_4=9$, $n_5=12$, and $n_k\le k!$ for $k\ge3$. #1064, not named on the page: B42 "Behavior of $\phi(\sigma(n))$ and $\sigma(\phi(n))$", p. 150 (PDF p. 168, text layer), Erdős's problem of proving that $\phi(n)>\phi(n-\phi(n))$ for almost all $n$ while $\phi(n)<\phi(n-\phi(n))$ for infinitely many $n$. #1065, not named on the page: no passage on primes of the form $2^kq+1$ for general $k$ or $2^k3^lq+1$ with $q$ prime was found in the text layer; the nearest passages are A7 "Cunningham chains", p. 30 (PDF p. 49, page image), which says that the Sophie Germain primes, primes $q$ with $2q+1$ also prime, are believed but not known to be infinitely many (the case $k=1$ of the page's first question), and A18, p. 66 (PDF p. 85), the remark that replacing $p+1$ by $p-1$ gives a similar classification, with its class tables and the question whether corresponding classes are equally dense. #1072, named on the page (through the site's remark): A2, p. 12 (PDF p. 31, page image), Hardy and Subbarao's belief that, with $f(p)$ "the least integer for which $f(p)!+1\equiv0\pmod p$", there are infinitely many $p$ with $f(p)=p-1$ but that the number of such $p\le x$ is $o(x/\ln x)$, and Erdős's belief "that $f(p)/p\to0$ for almost all $p$". #1074, not named on the page: A2, p. 12 (PDF p. 31, text layer), the Pillai primes, defined by Subbarao following Erdős as the primes $p$ for which some $n$ has $n!+1\equiv0\pmod p$ but $p\not\equiv1\pmod n$; those below 100 are 23, 29, 59, 61, 67, 71, 79 and 83; G. E. Hardy and Subbarao proved the Pillai primes infinite, along with the associated $n$, which they call EHS numbers, the first few being 8, 9, 13, 14, 15, 16, 17, 18, 19, 22; the question whether the Pillai primes have an asymptotic density, with experiments suggesting one between 0.5 and 0.6, and the corresponding density question for the EHS numbers. #1094, not named on the page: B31, p. 130 (PDF p. 148, text layer), the observation that most $\binom nk$ with $n\ge2k$ have a prime factor $p\le n/k$, Selfridge's conjecture, after computing with Lacampagne and Erdős, that this holds whenever $n>17.125k$, and the slightly stronger conjecture that every such coefficient has least prime factor at most $n/k$ or at most 17, with exactly 4 exceptions (their least prime factors 19, 19, 23 and 29), with the deficiency of a binomial coefficient and the Erdős--Selfridge function $g(k)$. #1113, not named on the page: B21 "$k\cdot2^n+1$ composite for all $n$", pp. 119--121 (PDF pp. 137--139, text layer), Sierpiński's covering-congruence construction of infinitely many such $k$, Selfridge's $78557$ with one of 3, 5, 7, 13, 19, 37, 73 always dividing $78557\cdot2^n+1$, the twelve remaining candidates below it, and Riesel's $k\cdot2^n-1$ with Stanton's six covering sets of divisors; the page's question, whether a Sierpiński number can have no finite covering set of primes, is not stated in the section. #1134, not named on the page: E36 "Klarner--Rado sequences", p. 361 (PDF p. 379, text layer), the sequence 1, 2, 4, 5, 8, 9, 10, 14, ..., the thinnest that contains 1 and, with each $x$, also $2x$, $3x+2$ and $6x+3$, and the question whether it has positive density, with the Klarner--Rado papers; the page's set is generated by $2x+1$, $3x+1$ and $6x+1$, and no equivalence with the printed generators is claimed here. #1135, named on the page (E16): E16 "The $3x+1$ problem", p. 330 (PDF p. 348, page image), Collatz's question, asked as a student, whether the sequence with $a_{n+1}=a_n/2$ for even $a_n$ and $a_{n+1}=3a_n+1$ for odd $a_n$ is tree-like apart from the cycle 4, 2, 1, 4, ... (Figure 16), that is, whether every starting integer $a_1$ reaches some $a_n=1$; the section quotes Erdős, "Mathematics may not be ready for such problems", and advises reading Jeff Lagarias's 1985 article or Eric Roosendaal's "On the $3x+1$ problem" page before attempting it; the site's "may not be" wording is Guy's, printed without date, occasion or source; the section continues with the September 2003 record, Eliahou's cycle-length criterion, Halbeisen--Hungerbühler and Oliveira e Silva's verification below $3\cdot2^{50}$ and cycle length at least 102225496, Krasikov--Lagarias's $x^{0.84}$, Bernstein's 2-adic reformulation and Crandall's $qx+1$ conjecture. #1142, not named on the page: A19, p. 67 (PDF p. 86, text layer), Erdős's conjecture that 4, 7, 15, 21, 45, 75 and 105 are the only $n$ for which $n-2^k$ is prime for every $k$ with $2\le2^k<n$, Mientka and Weitzenkamp's verification for $n<2^{44}$ and Uchiyama and Yorinaga's to $2^{77}$, Vaughan's bound of fewer than $x\exp\{-(\ln x)^c\}$ such numbers below $x$, short of $x^{1-\epsilon}$, with Hooley's conditional $A(x)=O(x^c)$ and Narkiewicz's improvement.

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