已知 (8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2) = p／q（p,q 为互质正整数），求 p+q

wintex

已知 (8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2) = p／q（p,q 为互质正整数），求 p+q

时空伴随者

\(\frac{p}{q}=\frac{1}{2(8-1)}\\p+q=1+2(8-1)=15\)

luyuanhong

题  已知 (8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2) = p／q（p,q 为互质正整数），求 p+q 。

解  p／q = (8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2)

    = (8+1)(8^2+1)(8^4+1)(8^8+1)／[(2^48-1)×2]

    = (8+1)(8^2+1)(8^4+1)(8^8+1)／[(8^16-1)×2]

    = (8+1)(8^2+1)(8^4+1)(8^8+1)／[(8^8+1)(8^8-1)×2]

    = (8+1)(8^2+1)(8^4+1)／[(8^8-1)×2]

    = (8+1)(8^2+1)(8^4+1)／[(8^4+1)(8^4-1)×2]

    = (8+1)(8^2+1)／[(8^4-1)×2]

    = (8+1)(8^2+1)／[(8^2+1)(8^2-1)×2]

    = (8+1)／[(8^2-1)×2]

    = (8+1)／[(8+1)(8-1)×2]

    = 1／[(8-1)×2]

    = 1／14 。

可见有

     p = 1 ，q = 14 ，p+q = 1+14 = 15 。

波斯猫猫

题：已知 (8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2) = p／q（p,q 为互质正整数），求 p+q 。

2^49-2=2(2^48-1)=2(2^24+1)(2^12+1)(2^12-1)=2(2^24+1)(2^12+1)(2^6+1)(2^6-1)

=2(2^24+1)(2^12+1)(2^6+1)(2^3+1)(2^3-1)=14(8^8+1)(8^4+1)(8^2+1)(8+1)，

p／q=(8+1)(8^2+1)(8^4+1)(8^8+1)／(2^49-2)=1/14，  P+q=15。