This is perhaps the first such integer: 1125899940397056. For this input, the answer should be 1125899906842624, but it gives 1125899973951489.

And the total number of such safe integers is tens of millions.

"Square it and you'll get the nearest square number."

Square it and you'll get the square of the nearest natural number - not necessarily the nearest square number.
They are different things and not actually the same.

THIS IS NOT NECESSARILY CORRECT

For an input 12.4, the answer should be 9, but this gives 16.

I know the question asks for integers, but there must be some integers for which this holds also.

I noticed that the random tests didn't seem to understand that A2345 is a valid straight. Probably 50-60% of my code is to ensure that worked with both flushed and straight flushes.

Note that K♠ Q♠ J♠ 10♣ 9♠ - 8♠ 7♠ is both a straight and a flush, but not a straight flush. This gets surprisingly bloody complicated when working with the high-low ace!