On sums of four pentagonal numbers with coefficients

The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2
(n=0,1,2,\ldots)$\mathrm{.}Let$(b,c,d)$beoneofthetriples$(1,1,2),(1,2,3),(1,2,6)$and$(2,3,4)$\mathrm{.}Weshowthateach$n=0,1,2,\ldots$ 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.