By Haskell Curry

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An excellent survey volume is Manin and Panchishkin, Introduction to Number Theory [96]. The best history is Weil, Number Theory: An Approach through History. From Hammurapi to Legendre [153]. There is also Leonard Eugene Dickson’s encyclopedic but unreadable three-volume History of the Theory of Numbers [25]. Guy’s Unsolved Problems in Number Theory [45] is a nice survey of unusual problems and results in elementary number theory. 16, see Nathanson [101]. Lang’s Algebra [89] is the standard reference for the algebra used in this book.

Usually the function f (x1 , . . , xk ) is a polynomial with rational or integer coeﬃcients. In this section we consider the linear diophantine equation a1 x1 + · · · + ak xk = b. We want to know when this equation has a solution in integers, and when it has a solution in nonnegative integers. For example, the equation 3x1 + 5x2 = b 38 1. Divisibility and Primes has a solution in integers for every integer b, and a solution in nonnegative integers for b = 0, 3, 5, 6, and all b ≥ 8 (Exercise 1).

We deﬁne the ﬁnite simple continued fraction a0 , a1 , . . , aN = a0 + 1 a1 + . 1 a2 + 1 .. 1 aN −1 + 1 aN Another notation for a continued fraction is 1 1 1 a0 , a1 , . . , aN = a0 + . ··· a1 + a2 + aN The numbers a0 , a1 , . . , aN are called the partial quotients of the continued fraction. For example, 2, 1, 1, 2 = 2 + 1 1+ 1 1+ 12 = 13 . 5 We can write a ﬁnite simple continued fraction as a rational function in the variables a0 , a1 , . . , aN . For example, a0 = a0 , 20 1. Divisibility and Primes a0 , a1 = and a0 , a1 , a2 = a0 a1 + 1 , a1 a0 a1 a2 + a0 + a2 .