19/06/02 18:34:50.16 OZg39pLw.net
>>402
問題５(1)
(b)
a_(k+i) = A_i と略記する。
A_4 = a_(k+4) = N,
A_1･A_3 + (A_2)^2 = A_0･N,
A_1･A_5 = (A_3)^2 + A_2･N,
A_1･A_2･A_6 = A_1(A_3･A_5 + NN) = A_3(A_1･A_5) + A_1･NN = (A_3)^3 + A_2･A_3･N + A_1･NN,
(A_1)^2･A_3･A_7 = (A_1)^2･{(A_5)^2 + A_6･N} = (A_1･A_5)^2 + (A_1)^2･A_6･N,
(c)
(A_1)^3･(A_2)^2･A_3 M
= (A_1)^3･(A_2)^2･A_3 {A_5･A_7 + (A_6)^2}
= (A2)^2･(A1･A5){(A1)^2･A3･A7} + A1･A3 (A1･A2･A6)^2
= (A2)^2･(A1･A5){(A1･A5)^2 + (A1)^2･A6･N} + A1･A3 (A1･A2･A6)^2
= (A2)^2･{(A3)^2 + A2･N}{[(A3)^2 + A2･N]^2 + (A1)^2･A6･N} + A1･A3 {(A3)^3 + A2･A3･N + A1･NN}^2
= (A3)^6･{A3･A1 + (A2)^2}
+ {(A1)^2･A2･A6 + 2A1･(A3)^3 +3(A2)^2･(A3)^2}A2･(A3)^2 N
+ {(A1)^2･(A2)^3･A6 + 3(A2)^4･(A3)^2 +2(A1)^2･(A3)^4 + A1･(A2)^2･(A3)^3} NN
+ {2(A1)^2･(A3)^2 + (A2)^4} A2 N^3
+ (A1)^3･A3 N^4
≡ (A_3)^6･{A_3･A_1 + (A_2)^2}
= A_0･(A_3)^6 N
≡ 0 (mod N)
ところで、(a) より A_1･A_2･A_3 と N=A4 は互いに素。
M は N の倍数。
A_8 = a_(k+8) = M/N は整数。