(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)等于多少?
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![(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)等于多少?](/uploads/image/z/344982-30-2.jpg?t=%282%5E1%2B1%29%282%5E2%2B1%29%282%5E4%2B1%29%282%5E8%2B1%29%282%5E16%2B1%29%282%5E32%2B1%29%282%5E64%2B1%29%E7%AD%89%E4%BA%8E%E5%A4%9A%E5%B0%91%3F)
(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)等于多少?
(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)等于多少?
(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)等于多少?
思路:在原式乘上(2-1),不断的产生平方差,可以巧解.
(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
原式=(2-1)(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^2-1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^4-1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^8-1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^16-1)(2^16+1)(2^32+1)(2^64+1)
=(2^32-1)(2^32+1)(2^64+1)
=(2^64-1)(2^64+1)
=2^128-1
(2^1+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^2-1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^4-1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
=(2^8-1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)
.
.
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=(2^64-1)(2^64+1)
=2^128-1
乘以(2-1)
原式=2^128-1