We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2\simeq A_1^2$, we show that penta-box has an of $D_3\simeq A_3$ provide strong evidence double-penta can be identified with a $D_4$ algebra. relate letters ${\bf u}$ variables...

Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea convert SLP corresponding Staircase Problem. main result (Theorem 1) shows that this works fine and results can be obtained by just calculating rational solutions an algebraic equation. Some examples such concrete calculations are given also given. In particular, w...