On universal sums of polygonal numbers

Details On universal sums of polygonal numbers On universal sums of polygonal numbers

2009-05-05

arxiv.org/abs/0905.0635v18

Mathematics

Number Theory

49 pages, typos corrected

Scientific paper

For m=3,4,..., the polygonal numbers of order m are given by $p_m(n)=(m-2)n(n-1)/2+n (n=0,1,2,...)$. For positive integers $a,b,c$ and $i,j,k>2$ with max{i,j,k}>4, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any n=0,1,2,... there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including (p_3,4p_4,p_5), (p_4,p_5,p_6) and (p_4,p_4,p_5)), we prove that any nonnegative integer can be written in the form $ap_i(x)+bp_j(y)+cp_k(z)$ with $x,y,z\in\Z$. We also show some related new results on ternary quadratic forms, one of which states that any nonnegative integer n=1(mod 6) can be written in the form $x^2+3y^2+24z^2$ with $x,y,z\in\Z$. In addition, we pose several related conjectures one of which states that for any m=3,4,... each natural number can be written as p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers and r among 0,...,m-3.

Affiliated with

Also associated with

No associations

Rating

On universal sums of polygonal numbers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On universal sums of polygonal numbers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On universal sums of polygonal numbers will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-450901