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Problem 44


Problem 44


Pentagon numbers

Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?


五边形数

五边形数由公式Pn=n(3n−1)/2生成。前十个五边形数是:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

可以看出P4 + P7 = 22 + 70 = 92 = P8。然而,它们的差70 − 22 = 48并不是五边形数。

在所有和差均为五边形数的五边形数对Pj和Pk中,找出使D = |Pk − Pj|最小的一对;此时D的值是多少?