Local obstructions and certified computations for perfect powers in power sums

Source abstract

Let $S_k(x)=1^k+\cdots+x^k$. We study perfect-power values of $S_k(x)$ by combining local valuations with finite congruence certificates. For an irregular Bernoulli pair $(p,r)$ with $e=v_p(B_r)\ge1$, we prove a valuation formula uniform for $k\equiv r+1\pmod{p^e(p-1)}$, subject to an explicit condition on $v_p(x(x+1))$. This gives explicit families of nonexistence results in which both $x$ and $k$ are unbounded. We also determine all solutions of $S_k(x)=y^n$ for $3\le x\le10^6$ with $x\equiv0,3\pmod4$, leaving $k$ and $n\ge2$ unrestricted and including squares. This extends Hajdu's fixed-$x$ classification in the same residue classes. A uniform restriction on the normalized odd part of $S_k(x)$ and finite certificates exclude every $k\ge6$; the remaining exponents are checked by exact integer arithmetic. The certificates admit independent verification. We further prove a limitation of congruence tests of bounded prime-power depth and solve the equation for every $x\equiv28\pmod{5\,110\,664}$.