2015/11/02 by Min Wang, Wang, Min, Zhi-Hong Sun +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Primary 11D85 #Secondary 11E25 #math.NT #msc:11D85 #msc:11E25
paper · pdf · doi:10.48550/arxiv.1511.00478
22 pages
arxiv created 2015/12/08 · arxiv updated 2015/12/09
Let \Bbb Z and \Bbb N be the set of integers and the set of positive integers, respectively. For a,b,c,d,n∈\Bbb N let N(a,b,c,d;n) be the number of representations of n by ax2+by2+cz2+dw2, and let t(a,b,c,d;n) be the number of representations of n by ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2 (x,y,z,w∈\Bbb Z). In this paper we reveal the connections between t(a,b,c,d;n) and N(a,b,c,d;n). Suppose a,n∈\Bbb N and 2\nmid a. We show that t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4) for (a,b,c,d)= (a,a,2a,8m), (a,3a,8k+2,8m+6), (a,3a,8m+4,8m+4) (n≡ m+\fraca-12 \pmod 2) and (a,3a,16k+4,16m+4) (n≡ \fraca-12\pmod 2). We also obtain explicit formulas for t(a,b,c,d;n) in the cases (a,b,c,d)=(1,1,2,8), (1,1,2,16),(1,2,3,6), (1,3,4,12), (1,1, 3,4), (1,1,5,5), (1,5,5,5), (1,3,3,12), (1,1,1,12), (1,1,3,12) and (1,3,3,4).