I suspect that there are an infinite number of such p^2 of the form

AbA, but that's strictly based on a simple heuristic argument.

The next two after your listed group are:

208139^2 = 43321843321

252717253^2 = 63866009963866009

Some experimentation for such numbers of the form n^2 seems to imply

that n^2 is perhaps more likely to be of the form AbA when the number

of digits in n is a multiple of 3. I can't really explain why, but I

haven't spent a lot of time thinking about it.

Statistics for the number of n with n^2 of form AbA:

n = 2 digits, solutions = 3

n = 3 digits, solutions = 4

n = 4 digits, solutions = 0

n = 5 digits, solutions = 2

n = 6 digits, solutions = 3

n = 7 digits, solutions = 1

n = 8 digits, solutions = 0

n = 9 digits, solutions = 21

On 8/28/2012 12:23 PM, woodhodgson@... wrote:

> Considering squares with an odd number of digits, the cases where the squared integers are prime and the squares less than 10^9 appear to be only these (the repetition meaning the square looks like AbA, where A is a string of digits and b any digit):

>

> 121 = 11^2, 29929 = 173^2, 69169 = 263^2, 732947329 = 27073^2.

>

> Does anyone know how uncommon these are for larger squares?

>

> {Regarding the related question about squares with an even number of digits and a complete repetition AA - I couldn't find any solutions at all for "small" squares, whether or not the original numbers were prime, and suspect there may not be any at all}.

>

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