We know that 2025 = 45
2. Therefore, 2025 is a perfect square in base ten.
Determine all positive integer value of N greater than 3, for which 2013 base N is a perfect square.
Provide valid reasoning for your answer.
Express 2013 base-N as a polynomial and factor.
(N+1)(2N^2-2N+3) which equals K^2.
If the first factor divides the second it evenly divides (2N^2+2N-4N-4+7), or (N+1) divides 7.
Then (N,K)=(6,21) is a solution since 2*6*6*6+6+3=441=21^2.
Otherwise we can assume no common factor >1, as we could divide it out.
So each factor will be a square, but 2N^2-2N+3 = 2N(N-1)+3 is of the form 4a+3 and can never be a square.
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Posted by xdog
on 2023-06-26 08:12:21 |
solution
