# Book:Richard K. Guy/Unsolved Problems in Number Theory/Third Edition

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## Richard K. Guy:

## Contents

## Richard K. Guy: *Unsolved Problems in Number Theory (3rd Edition)*

Published $\text {2004}$, **Springer-Verlag**

- ISBN 0-387-20860-7.

### Subject Matter

### Contents

- Preface to the Third Edition

- Preface to the Second Edition

- Preface to the First Edition

- Glossary of Symbols

- Introduction

- $\mathbf A$. Prime Numbers
- $\mathbf {A 1}$. Prime values of quadratic functions.
- $\mathbf {A 2}$. Primes connected with factorials.
- $\mathbf {A 3}$. Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.
- $\mathbf {A 4}$. The prime number race.
- $\mathbf {A 5}$. Arithmetic progressions of primes.
- $\mathbf {A 6}$. Consecutive primes in A.P.
- $\mathbf {A 7}$. Cunningham chains.
- $\mathbf {A 8}$. Gaps between primes.
- $\mathbf {A 9}$. Patterns of primes.
- $\mathbf {A 10}$. Gilbreath's conjecture.
- $\mathbf {A 11}$. Increasing and decreasing gaps.
- $\mathbf {A 12}$. Pseudoprimes.
- $\mathbf {A 13}$. Carmichael numbers.
- $\mathbf {A 14}$. "Good" primes and the prime number graph.
- $\mathbf {A 15}$. Congruent products of consecutive numbers.
- $\mathbf {A 16}$. Gaussian and Eisenstein-Jacobi primes.
- $\mathbf {A 17}$. Formulas for primes.
- $\mathbf {A 18}$. The Erdős-Selfridge classification of primes.
- $\mathbf {A 19}$. Values of $n$ making $n - 2^k$ prime. Odd numbers not of the form $\pm p^a \pm 2^b$.
- $\mathbf {A 20}$. Symmetric and asymmetric primes.

- $\mathbf B$. Divisibility
- $\mathbf {B 1}$. Perfect numbers.
- $\mathbf {B 2}$. Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers.
- $\mathbf {B 3}$. Unitary perfect numbers.
- $\mathbf {B 4}$. Amicable numbers.
- $\mathbf {B 5}$. Quasi-amicable or betrothed numbers.
- $\mathbf {B 6}$. Aliquot sequences.
- $\mathbf {B 7}$. Aliquot cycles. Sociable numbers.
- $\mathbf {B 8}$. Unitary aliquot sequences.
- $\mathbf {B 9}$. Superperfect numbers.
- $\mathbf {B 10}$. Untouchable numbers.
- $\mathbf {B 11}$. Solutions of $m \map \sigma m = n \map \sigma n$.
- $\mathbf {B 12}$. Analogs with $\map d n$, $\map {\sigma_k} n$.
- $\mathbf {B 13}$. Solutions of $\map \sigma n = \map \sigma {n + 1}$.
- $\mathbf {B 14}$. Some irrational series.
- $\mathbf {B 15}$. Solutions of $\map \sigma q + \map \sigma r = \map \sigma {q + r}$.
- $\mathbf {B 16}$. Powerful numbers. Squarefree numbers.
- $\mathbf {B 17}$. Exponential-perfect numbers.
- $\mathbf {B 18}$. Solutions of $\map d n = \map d {n + 1}$.
- $\mathbf {B 19}$. $\tuple {m, n + 1}$ and $\tuple {m + 1, n}$ with same set of prime factors.
- $\mathbf {B 20}$. Cullen and Woodall numbers.
- $\mathbf {B 21}$. $k \cdot 2^n + 1$ composite for all $n$.
- $\mathbf {B 22}$. Factorial $n$ as the product of $n$ large factors.
- $\mathbf {B 23}$. Equal products of factorials.
- $\mathbf {B 24}$. The largest set with no member dividing two others.
- $\mathbf {B 25}$. Equal sums of geometric progressions with prime ratios.
- $\mathbf {B 26}$. Densest set with no $l$ pairwise coprime.
- $\mathbf {B 27}$. The number of prime factors of $n + k$ which don't divide $n + i, 0 \le i < k$.
- $\mathbf {B 28}$. Consecutive numbers with distinct prime factors.
- $\mathbf {B 29}$. Is $x$ determined by the prime factors of $x + 1, x + 2, \dotsc, x + k$?
- $\mathbf {B 30}$. A small set whose product is square.
- $\mathbf {B 31}$. Binomial coefficients.
- $\mathbf {B 32}$. Grimm's conjecture.
- $\mathbf {B 33}$. Largest divisor of a binomial coefficient.
- $\mathbf {B 34}$. If there's an $i$ such that $n - 1$ divides $\binom n k$.
- $\mathbf {B 35}$. Products of consecutive numbers with the same prime factors.
- $\mathbf {B 36}$. Euler's totient function.
- $\mathbf {B 37}$. Does $\map \phi n$ properly divide $n - 1$?
- $\mathbf {B 38}$. Solutions of $\map \phi m = \map \sigma n$.
- $\mathbf {B 39}$. Carmichael's conjecture.
- $\mathbf {B 40}$. Gaps between totatives.
- $\mathbf {B 41}$. Iterations of $\phi$ and $\sigma$
- $\mathbf {B 42}$. Behavior of $\map \phi {\map \sigma n}$ and $\map \sigma {\map \phi n}$.
- $\mathbf {B 43}$. Alternating sums of factorials.
- $\mathbf {B 44}$. Sums of factorials.
- $\mathbf {B 45}$. Euler numbers.
- $\mathbf {B 46}$. The largest prime factor of $n$.
- $\mathbf {B 47}$. When does $2^a - 2^b$ divide $n^a - n^b$?
- $\mathbf {B 48}$. Products taken over primes.
- $\mathbf {B 49}$. Smith numbers.
- $\mathbf {B 50}$. Ruth-Aaron numbers.

- $\mathbf C$. Additive Number Theory
- $\mathbf {C 1}$. Goldbach's conjecture.
- $\mathbf {C 2}$. Sums of consecutive primes.
- $\mathbf {C 3}$. Lucky numbers.
- $\mathbf {C 4}$. Ulam numbers.
- $\mathbf {C 5}$. Sums determining members of a set.
- $\mathbf {C 6}$. Addition chains. Brauer chains. Hansen chains.
- $\mathbf {C 7}$. The money-changing problem.
- $\mathbf {C 8}$. Sets with distinct sums of subsets.
- $\mathbf {C 9}$. Packing sums of pairs.
- $\mathbf {C 10}$. Modular difference sets and error correcting codes.
- $\mathbf {C 11}$. Three subsets with distinct sums.
- $\mathbf {C 12}$. The postage stamp problem.
- $\mathbf {C 13}$. The corresponding modular covering problem. Harmonious labelling of graphs.
- $\mathbf {C 14}$. Maximal sum-free sets.
- $\mathbf {C 15}$. Maximal zero-sum-free sets.
- $\mathbf {C 16}$. Nonaveraging sets.
- $\mathbf {C 17}$. The minimum overlap problems.
- $\mathbf {C 18}$. The $n$ queens problem.
- $\mathbf {C 19}$. Is a weakly independent sequence the finite union of strongly independent ones?
- $\mathbf {C 20}$. Sums of squares.
- $\mathbf {C 21}$. Sums of higher powers.

- $\mathbf D$. Diophantine Equations
- $\mathbf {D 1}$. Sums of like powers. Euler's conjecture.
- $\mathbf {D 2}$. The Fermat problem.
- $\mathbf {D 3}$. Figurate numbers.
- $\mathbf {D 4}$. Waring's problem. Sums of $l$ $k$th Powers.
- $\mathbf {D 5}$. Sum of four cubes.
- $\mathbf {D 6}$. An elementary solution of $x^2 = 2 y^4 - 1$.
- $\mathbf {D 7}$. Sum of consecutive powers made a power.
- $\mathbf {D 8}$. A pyramidal diophantine equation.
- $\mathbf {D 9}$. Catalan conjecture. Difference of two powers.
- $\mathbf {D 10}$. Exponential diophantine equations.
- $\mathbf {D 11}$. Egyptian fractions.
- $\mathbf {D 12}$. Markoff numbers.
- $\mathbf {D 13}$. The equation $x^x y^y = z^z$
- $\mathbf {D 14}$. $a_i + b_j$ made squares
- $\mathbf {D 15}$. Numbers whose sums in pairs make squares.
- $\mathbf {D 16}$. Triples with the same sum and same product.
- $\mathbf {D 17}$. Product of blocks of consecutive integers not a power.
- $\mathbf {D 18}$. Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square.
- $\mathbf {D 19}$. Rational distances from the corners of a square.
- $\mathbf {D 20}$. Six general points at rational distances.
- $\mathbf {D 21}$. Triangles with integer edges, medians and area.
- $\mathbf {D 22}$. Simplexes with rational contents.
- $\mathbf {D 23}$. Some quartic equations.
- $\mathbf {D 24}$. Sum equals product.
- $\mathbf {D 25}$. Equations involving factorial $n$.
- $\mathbf {D 26}$. Fibonacci numbers of various shapes.
- $\mathbf {D 27}$. Congruent numbers.
- $\mathbf {D 28}$. A reciprocal diophantine equation.
- $\mathbf {D 29}$. Diophantine $m$-tuples.

- $\mathbf E$. Sequences of Integers
- $\mathbf {E 1}$. A thin sequence with all numbers equal to a member plus a prime.
- $\mathbf {E 2}$. Density of a sequence with l.c.m. of each pair less than $x$.
- $\mathbf {E 3}$. Density of integers with two comparable divisors.
- $\mathbf {E 4}$. Sequence with no member dividing the product of $r$ others.
- $\mathbf {E 5}$. Sequence with members divisible by at least one of a given set.
- $\mathbf {E 6}$. Sequence with sums of pairs not members of a given sequence.
- $\mathbf {E 7}$. A series and a sequence involving primes.
- $\mathbf {E 8}$. Sequence with no sum of a pair a square.
- $\mathbf {E 9}$. Partitioning the integers into classes with numerous sums of squares.
- $\mathbf {E 10}$. Theorem of van der Waerden. Szemerédi's theorem. Partitioning the integers into classes; at least one contains an A.P.
- $\mathbf {E 11}$. Schur's problem. Partitioning integers into sum-free classes.
- $\mathbf {E 12}$. The modular version of Schur's problem.
- $\mathbf {E 13}$. Partitioning into strongly sum-free classes.
- $\mathbf {E 14}$. Rado's generalizations of van der Waerden's and Schur's problems.
- $\mathbf {E 15}$. A recursion of Göbel.
- $\mathbf {E 16}$. The $3 x + 1$ problem.
- $\mathbf {E 17}$. Permutation sequences.
- $\mathbf {E 18}$. Mahler's $Z$-numbers.
- $\mathbf {E 19}$. Are the integer parts of the powers of a fraction infinitely often prime?
- $\mathbf {E 20}$. Davenport-Schinzel sequences.
- $\mathbf {E 21}$. Thue-Morse sequences.
- $\mathbf {E 22}$. Cycles and sequences containing all permutations as subsequences.
- $\mathbf {E 23}$. Covering the integers with A.P.s
- $\mathbf {E 24}$. Irrationality sequences.
- $\mathbf {E 25}$. Golomb's self-histogramming sequence.
- $\mathbf {E 26}$. Epstein's Put-or-Take-a-Square game.
- $\mathbf {E 27}$. Max and mex sequences.
- $\mathbf {E 28}$. $B_2$-sequences. Mian-Chowla sequences.
- $\mathbf {E 29}$. Sequence with sums and products all in one of two classes.
- $\mathbf {E 30}$. MacMahon's prime numbers of measurement.
- $\mathbf {E 31}$. Three sequences of Hofstadter.
- $\mathbf {E 32}$. $B_2$-sequences from the greedy algorithm.
- $\mathbf {E 33}$. Sequences containing no monotone A.P.s.
- $\mathbf {E 34}$. Happy numbers.
- $\mathbf {E 35}$. The Kimberling shuffle.
- $\mathbf {E 36}$. Klarner-Rado sequences.
- $\mathbf {E 37}$. Mousetrap.
- $\mathbf {E 38}$. Odd sequences.

- $\mathbf F$. None of the Above
- $\mathbf {F 1}$. Gauß's lattice point problem.
- $\mathbf {F 2}$. Lattice points with distinct distances.
- $\mathbf {F 3}$. Lattice points, no four on a circle.
- $\mathbf {F 4}$. The no-three-in-a-line problem.
- $\mathbf {F 5}$. Quadratic residues. Schur's conjecture.
- $\mathbf {F 6}$. Patterns of quadratic residues.
- $\mathbf {F 7}$. A cubic analog of Bhaskara's equation.
- $\mathbf {F 8}$. Quadratic residues whose differences are quadratic residues.
- $\mathbf {F 9}$. Primitive roots.
- $\mathbf {F 10}$. Residues of powers of two.
- $\mathbf {F 11}$. Distribution of residues of factorials.
- $\mathbf {F 12}$. How often are a number and its inverse of opposite parity?
- $\mathbf {F 13}$. Covering systems of congruences.
- $\mathbf {F 14}$. Exact covering systems.
- $\mathbf {F 15}$. A problem of R. L. Graham.
- $\mathbf {F 16}$. Products of small prime powers dividing $n$.
- $\mathbf {F 17}$. Series associated with the $\zeta$-function.
- $\mathbf {F 18}$. Size of the set of sums and products of a set.
- $\mathbf {F 19}$. Partitions into distinct primes with maximum product.
- $\mathbf {F 20}$. Continued fractions.
- $\mathbf {F 21}$. All partial quotients of one or two.
- $\mathbf {F 22}$. Algebraic numbers with unbounded partial quotients.
- $\mathbf {F 23}$. Small differences between powers of $2$ and $3$.
- $\mathbf {F 24}$. Some decimal digital problems.
- $\mathbf {F 25}$. The persistence of a number.
- $\mathbf {F 26}$. Expressing numbers using just ones.
- $\mathbf {F 27}$. Mahler's generalization of Farey series.
- $\mathbf {F 28}$. A determinant of value one.
- $\mathbf {F 29}$. Two congruences, one of which is always solvable.
- $\mathbf {F 30}$. A polynomial whose sums of pairs of values are all distinct.
- $\mathbf {F 31}$. Miscellaneous digital problems.
- $\mathbf {F 32}$. Conway's RATS and palindromes.

- Index of Authors Cited

- General Index

## Further Editions

- 1981: Richard K. Guy:
*Unsolved Problems in Number Theory* - 1994: Richard K. Guy:
*Unsolved Problems in Number Theory*(2nd ed.)

## Errata

### Numbers $n$ whose Euler Phi value Divides $n + 1$

$\mathbf B$: Divisibility: $\mathbf {B 37}$: Does $\map \phi n$ properly divide $n - 1$?

*Lehmer gives $8$ solutions to $\map \phi n \mid n + 1$, namely $n = 2$, $n = 2^{2^k} - 1$ for $1 \le k \le 5$, $n = n_1 = 3 \cdot 5 \cdot 17 \cdot 353 \cdot 929$ and $n = n_1 \cdot 83623937$. [Note that $353 = 11 \cdot 2^5 + 1, 929 = 29 \cdot 2^5 + 1, 83623937 = 11 \cdot 29 \cdot 2^{18} + 1$ and $\paren {353 - 2^8} \paren {929 - 2^8} = 2^{16} - 2^8 + 1$.] This exhausts the solutions with less than seven factors. Victor Meally notes that $n = n_1 \cdot 83623937 \cdot 699296672132097$ would be a solution were the largest factor a prime, put Peter Borwein notes that this is divisible by $73$.*

### Triples with Same Sum and Same Product

$\mathbf D$: Diophantine Equations: $\mathbf {D 16}$: Triples with Same Sum and Same Product

*It may be of interest to ask for the smallest sums or products with each multiplicity. For example, for $4$ triples, J. G. Mauldon finds the smallest common sum to be $118$ ... and the smallest common product to be $25200$ ...*

## Source work progress

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