  The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\
(n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples
$(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can
be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first
conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a
sum of five pentagonal numbers two of which are equal; this refines a classical
result of Cauchy claimed by Fermat.
