User blog comment:Vel!/Formal logic challenge/@comment-10429372-20140710101014

Second one:

∀a∈N∃b∈Z+∃c∈Z+∃d∈Z+∃e∈Z+ : a = b^2+c^2+d^2+e^2

Note I use Z+ instead of N, so any of b,c,d,e can be zero.