# 250 problems in elementary number theory by Waclaw Sierpinski By Waclaw Sierpinski

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R). , wr(a), 'respectively. , KWr of K. (n): There is a one-to-one correspondence 8. Let-fbuj} be a fundamental sequence with respect to w(a), and let (bm]}i where nii < m2

So far, the inequalities (1,1) and (1,2) have only been proved if F(x) satisfies the conditions (i) and (ii) of § 3, • (i): F(x) has the exact degree m, and (ii): F(x) is relatively prime to f (x). However, both inequalities remain valid if only the following weaker conditions are imposed, (if): F(x) is at most of degree m, and (ii1): F(a) * 0. For let mf be the exact degree of F(x). Then m' < m, hence ci (mT ) ^ ci (m), c2 (m1 ) ^ c2 (m). The inequalities (1,1) and (1,2) corresponding to the degree mf are thus at least as strong as those corresponding to the degree m and imply the latter.

The set g is an ideal of Ig. For let a and b be in 0, and let c be any element of Ig, so that Then alg, big) ^, 30 LECTURES ON DIOPHANTTNE APPROXIMATIONS and hence a+b, a-b, and ac are likewise in 9 . By the identity (II), |f|g = gla|g. Hence it follows that a belongs to 9 if and only if | belongs to Ig. • & Thus 9 consists of all multiples a=a'g where a'elg. In the language of ideal theory, 9 is the principal ideal g=(g) of Ig. The elements of Ig and 9 may also be characterised as follows. Write the rational number a as a quotient a = — of two rational integers P and that are relatively prime.