2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152701A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}μ(d) (1-x^d)^{-k/d}/k$, where $μ$ is the Möbius function. Equivalently, the size of the search space in which $k$-fold Euler sums of level $l$ are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{n<k}{\lfloor(l+n-1)/2\rfloor\choose n}$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\leq44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\leq7$; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with $l\leq 7$ is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings.34 pages, LaTeXHigh Energy Physics - TheoryHigh Energy Physics - PhenomenologyQuantum Algebratext