Cubic Problem

2014-12-27 11:42 pm

Let n1, n2 be integers with n2>n1>1, prove or disprove that there exists a positive integer M such that (n1)^3+((n1)+1)^3+…+(n2)^3=M^3.

回答 (1)

用戶2053

2014-12-29 11:54 pm

✔ 最佳答案

3³ + 4³ + 5³ = 6³
⇒ n2 = 5 > n1 = 3 > 1 , M = 6.
6³ + 7³ + ... + 69³
= (1³ + 2³ + ... + 69³) - (1³ + 2³ + ... + 5³)
= (69 × 70 / 2)² - (5 × 6 / 2)²
= (2415 - 15) (2415 + 15)
= (2³ × 30 × 10) (3 × 30 × 3³)
= 2³ × 3³ × 30³
= 180³
⇒ n2 = 69 > n1 = 6 > 1 , M = 180.
213³ + 214³ + ... + 555³
= (1³ + 2³ + ... + 555³) - (1³ + 2³ + ... + 212³)
= (555 × 556 / 2)² - (212 × 213 / 2)²
= (154290 - 22578) (154290 + 22578)
= (4³ × 7³ × 6) (6² × 17³)
= 4³ × 7³ × 6³ × 17³
= 2856³
⇒ n2 = 555 > n1 = 213 > 1 , M = 2856.

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