a problem

[托斯卡尼艷陽下]

a,b為1~100間的整數, besides...

a^3=b^2+b+1

試求 a,b 之可能值!

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Post by 托斯卡尼艷陽下
a,b為1~100間的整數, besides...
a^3=b^2+b+1
試求 a,b 之可能值!

由於 b≦100, b^2+b+1≦10101

3
因此 a≦√10101≒21.6

又兩邊同乘4後減去3可得

(2b+1)^2 = 4a^3-3

於是只要找 ≦21 的整數中 立方的四倍減3是平方數者即可

以下列表

a 4a^3-3
1 1 =1^2
2 29
3 105
4 253
5 497
6 861
7 1369 =37^2
8 2045
9 2913
10 3997
11 5321
12 6909
13 8785
14 10973
15 13497
16 16381
17 19649
18 23325
19 27433
20 31997
21 37041

其中只有兩個是平方數

由此得到 a=1, b=0 (不合題意) 或 a=7, b=18

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