BdMO National Higher Secondary 2020 P2

[Mursalin]

কতগুলো পূর্ণসংখ্যা \(n\) আছে যেন \(1\leq n\leq 2020\) এবং \(n^n\) একটা পূর্ণবর্গ সংখ্যা?


How many integers \(n\) are there subject to the constraint that \(1\leq n\leq 2020\) and \(n^n\) is a perfect square?

[Mehrab4226]

It can easily be seen that all even $n$ has the property. And also all the odd perfect square $n$ possible. That would make $1010+22 = 1032$ ($22$ because $45^2=2025 >2020)$

Other values of $n$ won't work because in their prime power factorization there is a prime with an odd-numbered power and when we power odd n over n then the power of that prime still remains odd. Thus the number can never be a square