整數n的s進制表示為777,若n為某整數的四次方,問s最小值為何?
回答 (1)
2021-04-19 10:36 pm
n = 7s^2+7s+7 = m^4
7 為質數, 故
7 | m, 且 7^3 | s^2+s+1
設 s^2+s+1 = 7^3 k, 則
n = 7^4 k = m^4
所以, 存在正整數 p 使 k = p^4.
故
s^2 + s + 1 = 7^3 p^4
即 s(s+1) = 7^3 p^4 - 1.
試 p = 1,
s(s+1) = 7^3 - 1 = 342 = 18．19
故 s = 18 是所求.
7 為質數, 故
7 | m, 且 7^3 | s^2+s+1
設 s^2+s+1 = 7^3 k, 則
n = 7^4 k = m^4
所以, 存在正整數 p 使 k = p^4.
故
s^2 + s + 1 = 7^3 p^4
即 s(s+1) = 7^3 p^4 - 1.
試 p = 1,
s(s+1) = 7^3 - 1 = 342 = 18．19
故 s = 18 是所求.
收錄日期: 2021-05-04 00:44:53
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