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Glossary of Symbols Glossary of Symbols a1 ≡ a2(mod b) a1 congruent to a2, modulo b; a1 − a2 divisible by b C the field of complex number d(n) the number of (positive) divisors of n; σo(n) d|nddivides n; n is a multiple of d; there is an integer q such that dq = n d | nddoes not divide n e base of natural logarithms; 2.718281828459045 ... exp{} exponential function 2n Fn Fermat numbers: 2 +1 Or Fibonacci numbers f(x)=0(g(x)) f(x)/g(x) → 0asx →∞ f(x)=0(g(x)) there is a constant c such that |f(x)| <cg(x) i square root of −1; i2 = −1 ln x natural logarithm of x (m, n) GCD (greatest common divisor) of m and n; highest common factor of m and n [m, n] LCM (least common multiple) of m and n. Also, the block of consecutive integers, m, m +1,...n p Mp Mersenne numbers: 2 − 1 Glossary of Symbols 341 n! factorial n; 1 × 2 × 3 × ...× n n k n choose k; the binomial coefficient n!/k!(n − k)! p or (p/q) Legendre symbol, also fraction q panpa divides n, but pa+1 does not divide n pn the nth prime, p1 =2, p2 =3,p3 =5,... Q the field of rational numbers rk(n) least number of numbers not exceeding n, which must contain a k-term arithmetic progression x Gauss bracket or floor of x; greatest integer not greater than x x ceiling of x; last integer not less than x xn least positive (or nonnegative) remainder of x modulo n Z the ring of integers Zn the ring of integers, 0, 1, 2,...,n− 1 (modulo n) γ Euler’s constant; 0.577215664901532 ... π ratio of circumference of circle to diameter; 3.141592653589793 ... π(x) number of primes not exceeding x, also primitive polynomial π(x; a, b) number of primes not exceeding x and congruent to a, modulo b product 342 Glossary of Symbols σ(n) sum of divisors of n σk(n) sum of kth powers of divisors of n sum φ(n) Euler’s totient function; number of positive integers not exceeding n and prime to n ω complex cube root of 1, ω3 =1,ω =1, ω2 + ω +1=0 ω(n) number of distinct prime factors of n Ω(n) number of prime factors of n, counting repetitions References Chapter 1 1.1 T. M. Apostol: Introduction to Analytic Number Theory (Springer, Berlin, Heidelberg, New York 1976) 1.2 I. Asimov: Asimov on Numbers (Doubleday, Garden City, NY, 1977) 1.3 A. O. L. Atkin, B. J. Birch (eds.): Computers in Number Theory (Academic, London 1971) 1.4 E. R. Berlekamp, J. H. Conway, R. K. Guy: Winning Ways (Academic, Lon- don 1981) 1.5 W. Kaufmann-B¨uhler: Gauss. A Biographical Study (Springer, Berlin, Hei- delberg, New York 1981) 1.6 P. J. Davis: The Lore of Large Numbers (Random House, New York 1961) 1.7 L. E. Dickson: History of the Theory of Numbers, Vols. 1–3 (Chelsea, New York 1952) 1.8 U. Dudley: Elementary Number Theory (Freeman, San Francisco 1969) 1.9 C. F. Gauss: Disquisitiones Arithmeticae [English transl. by A. A. Clarke, Yale University Press, New Haven 1966] 1.10 W. Gellert, H. K¨ustner,M. Hellwich, H. K¨astner(eds.) The VNR Concise Encyclopedia of Mathematics (Van Nostrand Reinhold, New York 1977) 1.11 R. K. 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