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 A fun prime number (Posted on 2011-02-15)
Find a prime number such that every two consecutive digits are a square number.

 See The Solution Submitted by Math Man Rating: 5.0000 (1 votes)

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 solution (spoiler) | Comment 1 of 4

The only 2-digit squares are 16, 25, 36, 49, 64 and 81.

These can be chained together in two ways before no longer being able to add to the chain: 81649 and 3649, as there is no 2-digit square beginning with a 9 or ending with an 8 or a 3.  A choice had come up before the 649, as two squares end with 6: 16 and 36.

Obviously any prime subset of these strings must have at least three digits, as any two digits constitute a square rather than a prime, and of course it must end in an odd digit. Candidates are the two strings themselves, 81649 and 3649, and 1649 and 649. Of these, only 81649 is prime, and is the answer.

3649 = 41*89
1649 = 17*97
649 = 11*59
81649 is prime.

 Posted by Charlie on 2011-02-15 19:12:02

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