可由对角三元二次型表示的算术级数

ARITHMETIC PROGRESSIONS REPRESENTED BY DIAGONAL TERNARY QUADRATIC FORMS

设d>r≥0为整数.对于正整数a,b,c,如果算术级数{r+dn:n=0,1,2,...}中每一项可写成ax2+by2+cz2(其中x,y,z为整数),则称ax2+by2+cz2是(d,r)-通用的.在本文中,我们使用三元二次型理论来研究一些对角三元二次型的(d,r)-通用性(由L.Pehlivan和K.S.Williams,以及孙智伟所猜测).例如:我们证明了2x2+3y2+10z2是(8,5)-通用的,x2+3y2+8z2与x2+2y2+12z2既是(10,1)-通用的也是(10,9)-通用的,3x2+5y2+15z2是(15,8)-通用的.

Let d>r≥0 be integers.For positive integers a,b,c,if any term of the arithmetic progression{r+dn:n=0,1,2,...}can be written as ax2+by2+cz2 with x,y,z∈Z,then the form ax2+by2+cz2 is called(d,r)-universal.In this paper,via the theory of ternary quadratic forms we study the(d,r)-universality of some diagonal ternary quadratic forms conjectured by L.Pehlivan and K.S.Williams,and Z.-W.Sun.For example,we prove that 2x 2+3y 2+10z 2 is(8,5)-universal,x 2+3y 2+8z 2 and x 2+2y 2+12z 2 are(10,1)-universal and(10,9)-universal,and 3x 2+5y 2+15z 2 is(15,8)-universal.