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Problem 518

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Prime triples and geometric sequences

Let S(n) = a+b+c over all triples (a,b,c) such that:

• a, b, and c are prime numbers.
• a < b < c < n.
• a+1, b+1, and c+1 form a geometric sequence.

For example, S(100) = 1035 with the following triples:

(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17), (7, 23, 71), (11, 23, 47), (17, 23, 31), (17, 41, 97), (31, 47, 71), (71, 83, 97)

Find S(10⁸).

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素数三元组和等比数列

对于所有满足下列条件的三元组(a,b,c)，记S(n) = a+b+c之和：

• a、b、c均为素数。
• a < b < c < n。
• a+1、b+1、c+1构成等比数列。

例如，由于存在下列三元组，S(100) = 1035：

(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17), (7, 23, 71), (11, 23, 47), (17, 23, 31), (17, 41, 97), (31, 47, 71), (71, 83, 97)

求S(10⁸)。

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