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The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\\binom n2+n$ $(n=0,1,2,\\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. 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