Integral part and fractional part

Algebra Level 4

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

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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