Multi-loop scattering amplitudes are core ingredients in high-precision predictions for particle collider cross-sections. In cutting-edge amplitude calculations for modern colliders, one of the biggest challenges comes from loop integrals, whose manipulation leads to the appearance of very large rational functions of the kinematic invariants. The algebraic complexity associated with handling or obtaining these rational functions can often be the limiting factor in the calculation of higher-order amplitudes. In this talk I will discuss techniques employing p-adic numbers from number theory to help obtain, simplify, and study the rational functions in loop amplitude calculations. I will apply these techniques to state-of-the-art examples from 2-loop 5-point QCD calculations for the LHC.
