Integral part and fractional part

Algebra Level 4

On your calculator you have the number 4+7+5324.14005494464 4 + \frac{ -7 + \sqrt{53} } { 2} \approx 4.14005494464 and by mistake you hit the key that squares the number giving you 17+7+53217.1400549446417 + \frac{ -7 + \sqrt{53} } { 2} \approx 17.14005494464.

You notice that the whole number part of the number and the whole number part of the square of the number has changed (4 to 17) but that the decimal parts of both are still the same.

How many numbers strictly between 4 and 5 have that property? (That is, when you square the number, the whole number part changes but the decimal part remains the same.)

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