A conjecture regarding sums of consecutive powers Is it true that for any $n \ge 2$, there is a tuple of $n+1$ consecutive natural numbers such that the sum of the $n$th powers of the first $n$ natural numbers is equal to the $n$th power of the last natural number? I was led to this conjecture after seeing the pattern hold for $(3,4,5)$ and $(3,4,5,6)$. It does not hold for $(3,4,5,6,7)$, but I wonder if a length $5$ tuple exists.
 A: No such tuple of length $5$ exists. In particular, we can suppose that your tuple is of the form $(a,a+1,a+2,a+3,a+4)$ and satisfies:
$$a^4+(a+1)^4+(a+2)^4+(a+3)^4=(a+4)^4$$
or, equivalently
$$a^4+(a+1)^4+(a+2)^4+(a+3)^4-(a+4)^4=0$$
Now, this makes clear that we're just looking for the root to a polynomial in $a$. We can expand each of the products above (for instance, by asking Wolfram|Alpha) to get
$$3a^4 + 8a^3 - 12a^2 - 112 a -158=0$$
and then using the rational root theorem we can see that the only possible rational solutions $\frac{p}q$ have that $p$ is a factor of $158$ (i.e. one of $\pm 1,\,\pm 2,\,\pm 79,\pm 158$) and $q$ is a factor of $3$ (i.e. $1$ or $3$). Testing yields that none of these are roots, so no integral (nor rational) $a$ works to create such a tuple.
One may also plot this polynomial, and simply note that it doesn't have any roots at integers; knowing a little bit about the general behavior of polynomials (of low degree) tells us that nothing surprising will happen that a graph can't show us.
