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Glossary of Symbols Glossary of Symbols a1 ≡ a2(modb) 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 lnx natural logarithm of x (m,n) GCD (greatest common divisor) of m and n; highest common factor of m and n 405 406 Glossary of Symbols [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 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,butpa+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 Glossary of Symbols 407 π(x;a,b) number of primes not exceeding x and congruent to a, modulo b ∏ product 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, London 1981) 1.5 W. Kaufmann-Buhler:¨ Gauss. A Biographical Study (Springer, Berlin, Heidelberg, 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. Kustner,¨ M. Hellwich, H. Kastner¨ (eds.) The VNR Concise Encyclopedia of Mathematics (Van Nostrand Reinhold, New York 1977) 1.11 R. K. 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