1977 Canadian MO Problems/Problem 3

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$\displaystyle N$ is an integer whose representation in base $\displaystyle b$ is $\displaystyle 777.$ Find the smallest positive integer $\displaystyle b$ for which $\displaystyle N$ is the fourth power of an integer.


Rewriting $\displaystyle N$ in base $\displaystyle 10,$ $\displaystyle N=7(b^2+b+1)=a^4$ for some integer $\displaystyle a.$ Because $\displaystyle a^4|7$ and $\displaystyle 7$ is prime, $\displaystyle a \ge 7^4.$ Since we want to minimize $\displaystyle b,$ we check to see if $\displaystyle a=7^4$ works.

When $\displaystyle a=7^4,$ $\displaystyle b^2+b+1=7^3.$ Solving this quadratic, $\displaystyle b = 18$.

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