Guy, Richard K. Unsolved problems in number theory. 2nd ed. (English) Zbl 0805.11001 Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics 1. New York, NY: Springer-Verlag (ISBN 0-387-94289-0/hbk). xvi, 285 p. (1994). Since the appearance in 1981 of the first edition ( Zbl 0474.10001) [A Japanese translation with some corrections and additions was published in 1983 (see Zbl 0549.10001)] of this collection of open problems in number theory essential progress has been made: the Mordell conjecture has been proved, infinitely many Carmichael numbers have been found and possibly even the mystery of Fermat’s Last Theorem has been resolved. The present new edition takes account of these developments: the comments are updated and the bibliography essentially extended. Moreover several new problems have been added; there are seven new subsections dealing mostly with some new sequences of integers. Reviewer: W.Narkiewicz (Wrocław) Cited in 21 ReviewsCited in 202 Documents 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 Keywords:unsolved problems; prime numbers; divisibility; additive number theory; diophantine equations; bibliography; sequences of integers Citations:Zbl 0474.10001; Zbl 0549.10001 PDFBibTeX XMLCite \textit{R. K. Guy}, Unsolved problems in number theory. 2nd ed. New York, NY: Springer-Verlag (1994; Zbl 0805.11001) Online Encyclopedia of Integer Sequences: Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p. Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board. Smallest number m such that the trajectory of m under iteration of Euler’s totient function phi(n) [A000010] contains exactly n distinct numbers, including m and the fixed point. From Gilbreath’s conjecture. Numbers n such that there exists a solution to the equation 1 = 1/x_1 + ... + 1/x_k (for any k), 0 < x_1 < ... < x_k = n. Number of initial odd numbers in class n of the iterated phi function. Semiprime triangle, read by rows. 3-almost prime triangle, read by rows. Prime tetrahedron, read by rows. Numbers whose squares added to 7! are prime. Smallest natural d = d(n) such that 2^n + d is lesser of twin primes (n = 0, 1, 2, ...). Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square. a(n) = abs(2^n-127). Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908). Number of primes of the form k^2 + 1 less than n. Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1). Benoît Perichon’s 26 primes in arithmetic progression. Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.