Problems & Puzzles: Puzzles

Puzzle 1192 Follow up to Puzzle 258 Carlos Rivera propose explore the following puzzle as a follow-up to Puzzle 258 Phil Carmody found a prime p such that p^2 contains two successive concatenated integers. 811451682377384625400019885321^2 = 658453832833087902225991273042&658453832833087902225991273041 (60 digits) Q. Find a prime whose square or cube contains 3 successive concatenated integers (no matter the order)

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From 5-11 Oct 2024 contributions came from Emmanuel Vantieghem

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There are no primes whose square is a concatenation of three consecutive numbers

since any such concatenation is divisible by three.
Nevertheless, it can be interesting to search for squares that are such concatenations.
They must be very rare.  I only know of :
   1089 = 33^2
   576 = 24^2
   324 = 18^2.
Other examples must have at least  30 digits.

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