Problem:
Given positive integer $x$ find minimum value of positive integer $N$ starting which the following is always composite
$$ 1^x+2^x+3^x+4^x+\cdots+N^x $$
My Thoughts:
This is a followup to this problem
The original author of the problem wrongly mentioned the minimum condition only for $x=1$ and $x=2$ . But in fact for every $x$ there should be some minimum starting which it is always composite.
- For $x=1, N_{min}=2$
- For $x=2, N_{min}=3$
- For $x=3, N_{min}=2$
- For $x=4, N_{min}=3$
For $x=1440$ @resuns have found a small enough $N=5$ for which it breaks, it's obvious for $x=1440$ minimum $$ for making the series composite must be higher.
To understand this intuitively, if we look at Faulhaber's formulas, they are always expressed as factors(to be proved as well) with a constant in denominator. So if the product of the factors except the largest one is greater than the denominator then you will have some factor($\gt1$) left alongside the largest factor even after denominator gets cancelled out complete with the earlier factors.
Larger the $x$ is, larger is the minimum value of $N$ required to be for the series to be composite. I believe it should be possible to find an asymptotic if not an exact formula for it.
More Clarifications:
We might also need to prove that starting that minimum $N$ for any higher $N$ the series is composite. Or else it is not the real minimum value of $N$ but a mere false positive case. It should not be such that for a string of $N$s we get composite values then we encounter a prime value, in that case the last found minimum is wrong and the real minimum is higher or if you can prove such minimum can't exist you are welcome to write about it as well. I am not claiming we will get a run of primes followed by a run of composites and that will be end of it. We might even encounter a false positive minimum if start with such assumptions. I am only claiming there must be an infinite run of composites at the end which needs to be proved (or disproved) and if proved we need to find an asymptotic formula for it .