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Unsolved problems in number theory. 3rd ed. (English) Zbl 1058.11001

Problem Books in Mathematics. New York, NY: Springer-Verlag (ISBN 0-387-20860-7/hbk). xviii, 437 p. (2004).
This is the third edition of Guy’s well-known problem book on number theory (see Zbl 0474.10001 and Zbl 0805.11001 for previous editions), which is organised into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. Most of the problems are of an elementary, combinatorial nature. All of them can be understood without too much effort, but solving them is a different matter; indeed some of the problems are hopelessly difficult, including a few notorious ones. Being problems in number theory, even a complete solution to an original problem often generates yet more related problems. Thus many of the problems from the earlier editions have been expanded with more up to date comments and remarks. For most readers the bibliographies are the more useful aspects of the book; however, it has become so extensive for many of the problems that some pruning has become necessary. There are also several new problems, but the more interesting new feature is that, for about half of the problems, there is now a link of references to N. J. A. Sloane’s [Notices Am. Math. Soc. 50, No. 8, 912–915 (2003; Zbl 1044.11108)] very useful Online Encyclopedia of Integer Sequences.
As the author remarks, the book is perpetually out of date, but there are STOP PRESS entries. For example, in the problem concerning arithmetic progression of primes, reference is given for the recent announcement by Ben Green and Terence Tao that the primes contain arbitrarily long arithmetic progressions. There is little doubt that a new generation of talented young mathematicians will make very good use of this book, although it has to be said that the new edition can do with some copy-editing and proof-reading, because there are many easily detectable but irritating misprints.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
00A07 Problem books
11Bxx Sequences and sets
11Dxx Diophantine equations
11Nxx Multiplicative number theory
11Pxx Additive number theory; partitions
11Axx Elementary number theory

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Online Encyclopedia of Integer Sequences:

Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).
Noncototients: numbers k such that x - phi(x) = k has no solution.
Main diagonal of Kimberling’s expulsion array (A035486).
Indices of prime Mersenne numbers (A001348).
n + 1 - d(n) - phi(n) never takes these values (conjecture).
Primes or negative values of primes in the sequence b(n) = 47*n^2 - 1701*n + 10181, n >= 0.
Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes.
Numbers n such that n | Sum_{k=1..n} k!.
Value of x of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.
Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.
Conjectured values for a(n) = least natural number k such that phi(n+k) = phi(n) + phi(k), if k exists; otherwise 0.
Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.
a(n) = Max d(j), j=1..n-1, where d(j) is the smallest positive number such that 2j+d(j) and 2n+d(j) are both prime. A generalization of A073310.
Mersenne primes p such that M(p) = 2^p - 1 is also a (Mersenne) prime.
Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.
Conjectured values of n such that sigma(n)+sigma(k)=sigma(n+k) has no solution.
Least k such that phi(n+k)=2*phi(n), where phi is Euler’s totient function.
Primes p such that p^2 divides m!+1 for some integer m<p.
Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).
Conjectured number of numbers k such that sigma(k)/k = n.
Primes prime(j) which cannot be written as 2*prime(j) = prime(j+k) + prime(j-k) for any 0 < k < j.
Decimal expansion of Sum_{k>=1} phi(k)/2^k, where phi is Euler’s totient function.
Cardinality of the union of the set of sums and the set of products made from pairs of integers from {1..n}.
Smallest possible cardinality of the union of the set of pairwise sums and the set of pairwise products from a set of n positive integers.
a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n cubes, or 0 if no such magic square exists.
Number of primes between n and n+log(n)^2.
Binomial coefficients binomial(n,k) = UV such that n>=2k and U > V, where gpf(U) <= k, gpf(V) > k (gpf(n)= is the greatest prime factor of n).
Numbers n that give record values for s(n)*phi(n)/n^2, where s(n) is the sum of squares of the differences between consecutive totatives of n (A322144).
Counterexamples to a conjecture of Selfridge and Lacampagne.
Main diagonal of right-and-left variant of Kimberling expulsion array, A007063.