Erdős, Paul; Graham, Ronald L. Old and new problems and results in combinatorial number theory. (English) Zbl 0434.10001 Monographie No. 28 de L’Enseignement Mathématique. Genève: L’Enseignement Mathématique, Université de Genève. 128 p. (1980). This survey of open problems contains chapters on each of the following topics: van der Warden’s theorem and related topics, covering congruences, unit fractions, bases, completeness of sequences, irrationality and transcendence, diophantine problems, miscellaneous problems. There then follow ten pages of remarks on an earlier similar paper [P. Erdős, ibid. 6, 81–135 (1963; Zbl 0117.02901)], bringing that survey up to date. A full bibliography, and an author index of over 300 names, are also included. Fascinating to dip into. Reviewer: Ian Anderson (Glasgow) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 32 ReviewsCited in 188 Documents MSC: 11-02 Research exposition (monographs, survey articles) pertaining to number theory 00A07 Problem books 11A07 Congruences; primitive roots; residue systems 11B25 Arithmetic progressions 11B75 Other combinatorial number theory 11B83 Special sequences and polynomials 11D68 Rational numbers as sums of fractions 11J72 Irrationality; linear independence over a field 11J81 Transcendence (general theory) Keywords:combinatorial number theory; arithmetic progression; survey of open problems; van der Waerden’s theorem; covering congruences; unit fractions; bases; completeness of sequences; irrationality and transcendence; diophantine problems Citations:Zbl 0117.02901 × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists. Decimal expansion of sum of reciprocals of lcm(1..n) = A003418(n). Numbers of distinct integers obtained from summing up subsets of {1, 1/2, 1/3, ..., 1/n}. Let a(n) be the least k such that in the prime power factorization of k! the exponents of primes p_1, ...,p_n are even, while the exponent of p_(n+1) is odd. Let the prime factorization of (2*n)! be 2^e_1*3^e_2*5^e_3*...; then a(n) = maximal k such that e_1, ..., e_k are all even.. a(n) is the smallest k such that in the prime power factorization of k! at least the first n positive exponents are even. Smallest k such that the number of the first even exponents in prime power factorization of (2*k)! is n, or a(n)=0 if there is no such k. The 2-color Rado numbers for x_1^2 + x_2^2 + ... + x_n^2 = z^2. Decimal expansion of Sum_{k>=1} 1/(2^k - 3). Decimal expansion of Sum_{k>=2} 1/(k! - 1). Decimal expansion of Sum_{k >= 1} 2^(-phi(k)), where phi is the Euler totient function (A000010). Decimal expansion of Sum_{k >= 1} 2^(-sigma(k)), where sigma is the sum of divisors function (A000203). Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032). Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045).