A Paradox!!!

Number Theory Level 3

Let $n,k>1$ be a positive integer. Find the largest number $M$ and also the type of $n$ such that $n \mid 1^n+2^n+3^n+...n^k$ whenever $n<M$ . Let $S=1^n+2^n+3^n+...n^n$

No such restriction on n but

M \leq n^n

No restriction on

M

n

M= 23^2018+17^2019

,n is always such that

n^M \equiv 1 \mod M^n

M=17^2018+23^2019

, type of

n

n \equiv 0 \mod 4

and

0 \mod 8

No such restriction on M but no prime

p > k.n^{\frac{3}{2}}

can ever divide S for n>M

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A Paradox!!!

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