Number Theory: An Introduction to Mathematics
146 III More on Divisibility An idealA={ 0 }is said to bedivisibleby an idealB,andBis said to be afactor ofA, if there exists a ...
2 Quadratic Fields 147 x^2 −sx+rt= 0 with integer coefficientss=βγ′+β′γandrt=ββ′γγ′. It follows thatβγ′/land β′γ/lare roots of t ...
148 III More on Divisibility In the terminology of Chapter II,§1, this shows thatany two nonzero ideals inOd have a greatest com ...
2 Quadratic Fields 149 (i) pOd=PP′and P=P′; (ii) pOd=P=P′; (iii)pOd=P^2 and P=P′. Proof IfPis a prime ideal inOd,thenPP′ =lOdfo ...
150 III More on Divisibility Suppose next thatp|d.Thend=pefor somee∈Zandpe,sincedis square- free. IfP=(p, √ d),then P^2 =p(p, √ ...
2 Quadratic Fields 151 In the present case we must havea=2,c=1and b(b− 1 )≡(d− 1 )/4mod2 ifd≡1mod4. Sinceb(b− 1 )is even, it fol ...
152 III More on Divisibility 1993 Fermat’s assertion had been established in this way for allnless than four million. However, t ...
3 Multiplicative Functions 153 For any two functionsf,g:N→C, we define theirsum f+g:N→Cin the natural way: (f+g)(n)=f(n)+g(n). I ...
154 III More on Divisibility It follows from Lemma 25 that the set of all arithmetical functions f :N→C such thatf( 1 )=0 is an ...
3 Multiplicative Functions 155 h(n)= ∑ d′|n′,d′′|n′′ f(d′d′′)g(n′n′′/d′d′′) = ∑ d′|n′,d′′|n′′ f(d′)f(d′′)g(n′/d′)g(n′′/d′′) = ∑ ...
156 III More on Divisibility By Proposition II.24, Euler’sφ-function satisfiesi∗φ=j. Thusφ=i−^1 ∗j,and Propositions 26 and 27 pr ...
3 Multiplicative Functions 157 Proof Letf:N→Cbe given and putfˆ=f∗i.Then fˆ∗μ=f∗i∗μ=f∗δ=f. Conversely, letfˆ:N→Cbe given and put ...
158 III More on Divisibility The sufficiency of the condition in Proposition 30 was proved in Euclid’sElements (Book IX, Proposi ...
3 Multiplicative Functions 159 Suppose first thatMpdividesSp− 1 and assume thatMpis composite. Ifqis the least prime divisor ofM ...
160 III More on Divisibility We turn now from the primality of 2m−1 to the primality of 2m+1. It is easily seen that if 2m+1 is ...
4 Linear Diophantine Equations 161 Proposition 32If m> 1 ,thenN:= 2 m+ 1 is prime if and only if 3 (N−^1 )/^2 + 1 is divisibl ...
162 III More on Divisibility If a moduleMis generated by the elementsa 1 ,...,an, then it is also generated by the elementsb 1 , ...
4 Linear Diophantine Equations 163 common divisor ofa 2 ,...,an, there exists an invertible(n− 1 )×(n− 1 )matrixV′ such that (a ...
164 III More on Divisibility The elementsb 1 ,...,brof a moduleMare said to be abasisforMif they generate Mand are linearly inde ...
4 Linear Diophantine Equations 165 Supposem 1 =m 2 .Letaandbbe the entries in them 1 -th row of the first and second columns, an ...
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