Question

Let $n>1$ be a positive integer, then find the largest integer m such that ($n^m$ + 1) divides $1+n+n^2$ + ......+ $n^{127}$.

Answer 1

Since $n^m$ + 1 divides $1+n+n^2$ + ......+ $n^{127}$

Therefore $\frac{1+n+n^2+........+n^{127}}{n^m+1}$ is an integer

⇒ $\frac{1-n^{128}}{1-n}$ × $\frac{1}{n^m+1}$ is an integer

⇒ $\frac{(1-n^{64})(1+n^{64})}{(1-n)(n^m+1)}$

is an integer when m = 64.