Number Theory: An Introduction to Mathematics

166 III More on Divisibility By=c. But the latter system is soluble if and only ifcis an integral linear combination of the firs ...

4 Linear Diophantine Equations 167 A 1 X=A 2. We say in this case thatA 1 is aleft divisorofA 2 ,orthatA 2 is aright multipleofA ...

168 III More on Divisibility Proposition 39For any m×n matrix A, the following conditions are equivalent: (i)for some, and hence ...

4 Linear Diophantine Equations 169 thatb=y 1 b 1 +b′for somey 1 ∈ Rand someb′in the moduleM′generated by a 2 ,...,an.Thesetofall ...

170 III More on Divisibility Proposition 41Let R be a principal ideal domain and let A be an m×n matrix with entries from R. If ...

4 Linear Diophantine Equations 171 On the other hand, ifa 11 dividesa 1 kfor allk>1 then, by subtracting multiples of the fir ...

172 III More on Divisibility impliesxk =ykdkfor someyk ∈ Rif 1≤k≤randxk =0ifr <k ≤n.In particular,xka′k=Ofor 1≤k≤n, and thus ...

4 Linear Diophantine Equations 173 a 11 (D)x 1 +···+a 1 n(D)xn=c 1 (t) a 21 (D)x 1 +···+a 2 n(D)xn=c 2 (t) ··· am 1 (D)x 1 +···+ ...

174 III More on Divisibility 5 FurtherRemarks For the history of the law of quadratic reciprocity, see Frei [16]. The first two ...

5 Further Remarks 175 For the early history of Fermat’s last theorem, see Vandiver [52], Ribenboim [41] and Kummer [28]. Further ...

176 III More on Divisibility For proofs of these results and for later developments, see Lam [29], Fitchas and Galligo [14], and ...

6 Selected References 177 [18] S. Gelbart, An elementary introduction to the Langlands program,Bull. Amer. Math. Soc. (N.S.) 10 ...

178 III More on Divisibility [50] R.G. Swan, Gubeladze’s proof of Anderson’s conjecture,Azumaya algebras, actions and modules(ed ...

IV Continued Fractions and Their Uses............................ 1 TheContinuedFractionAlgorithm............................. L ...

180 IV Continued Fractions and Their Uses with matrix T′= ( α′ β′ γ′ δ′ ) , then, as is easily verified, the matrix T′′= ( α′′ β ...

1 The Continued Fraction Algorithm 181 ( pn pn− 1 qn qn− 1 )( 11 −an 0 ) = ( pn− 2 pn qn− 2 qn ) , from which, by taking determi ...

182 IV Continued Fractions and Their Uses wherebnandcnare integers andcn>0. We can write bn=ancn+cn+ 1 , wherean=ξnandcn+ 1 ...

1 The Continued Fraction Algorithm 183 Proof We h av e ηn+ 1 =(qn− 1 η−pn− 1 )/(pn−qnη). Hence θn+ 1 :=qnηn+ 1 +qn− 1 =(pnqn− 1 ...

184 IV Continued Fractions and Their Uses Sinceη>1, its continued fraction expansion has the form [an,an+ 1 ,...], where an≥1 ...

2 Diophantine Approximation 185 betweenξand its complete quotientξnit follows that η=(anξn+bn)/(cnξn+dn), where an=apn− 1 +bqn− ...