Universal sums of generalized octagonal numbers

作者：Ju Jangwon^*; Oh Byeong Kweon

来源：Journal of Number Theory, 2018, 190: 292-302.

DOI：10.1016/j.jnt.2017.12.014

• Lagrange's four square theorem
• Generalized octagonal numbers

An integer of the form P-8(x) = 3x(2) - 2x for some integer x is called a generalized octagonal number. A quaternary sum Phi(a,b,)(c,)(d)(x, y, z, t) = aP(8)(x) + bP(8)(y) + cP(8)(z) + dP(8)(t) of generalized octagonal numbers is called universal if Phi(a,b,)(c,)(d)(x, y, z, t) = n has an integer solution x,y,z,t for any positive integer n. In this article, we show that if a = 1 and (b,c,d) = (1,3,3), (1,3,6), (2,3,6), (2,3,7) or (2,3,9), then Phi(a,b,)(c,)(d)(x, y, z, t) is universal. These were conjectured by Sun in [10]. We also give an effective criterion on the universality of an arbitrary sum a(1)P(8)(Xi) a(2)P(8)(X2) +. . .+ a(k)P(8)(X-k) of generalized octagonal numbers, which is a generalization of "15-theorem" of Conway and Schneeberger.

• 出版日期2018-9