Status: exploratory, archived. Not an OEIS submission.
This folder documents the p = 5 member of the power-balance family: the analogue, for sums of four fifth powers, of the four-square balance-gap record locations A396760 (the p = 2 case, the active sequence of this repository). It is retained for reference and reproducibility. If you use or extend these terms, please reference this repository.
For a representable integer N, write
N = a^5 + b^5 + c^5 + d^5, 0 <= a <= b <= c <= d (integers).
Let minmax(N) be the least possible largest part d over all such representations, and let lb(N) be the least integer m with 4*m^5 >= N. The balance gap is
g(N) = minmax(N) - lb(N).
a(n) is the n-th representable N, taken in increasing order, at which g(N) exceeds all earlier values. Unlike p = 2 (Lagrange), for p >= 3 not every integer is a sum of four p-th powers; records are taken over the representable N only.
1, 3125, 59049, 371293
These 4 terms are complete for N <= 800000 (exhaustive scan); nothing is asserted beyond that bound. The corresponding record gaps are 0, 1, 2, 3. Within the scanned range every record location beyond 1 is itself a fifth power:
3125 = 5^5, 59049 = 9^5, 371293 = 13^5.
This is an observation about the scanned range only.
b-file.txt — the 4 verified terms, offset 1.program.gp — PARI/GP scanner: lista(Nmax) prints every record location <= Nmax.../../exploratory-src/balance_records.py — reference computation for the whole
p = 2..7 family.../../../viz/power-balance/ — interactive visualization of the family.