An integer of the form P 8 pxq " 3x 2´2 x for some integer x is called a generalized octagonal number. A quaternary sum Φ a,b,c,d px, y, z, tq " aP 8 pxq`bP 8 pyq`cP 8 pzq`dP 8 ptq of generalized octagonal numbers is called universal if Φ a,b,c,d px, y, z, tq " n has an integer solution x, y, z, t for any positive integer n. In this article, we show that if a " 1 and pb, c, dq " p1, 3, 3q, p1, 3, 6q, p2, 3, 6q, p2, 3, 7q or p2, 3, 9q, then Φ a,b,c,d px, y, z, tq is universal. These were conjectured by Sun in [11]. We also give an effective criterion on the universality of an arbitrary sum a 1 P 8 px 1 q`a 2 P 8 px 2 q`¨¨¨`a k P 8 px k q of generalized octagonal numbers , which is a generalization of "15-theorem" of Conway and Schneeberger.2000 Mathematics Subject Classification. Primary 11E12, 11E20.

An integer of the form [Formula: see text], for some integer [Formula: see text], is called a generalized polygonal number of order [Formula: see text]. A ternary sum [Formula: see text] of generalized polygonal numbers, for some positive integers [Formula: see text] and some integers [Formula: see text], is said to be universal over [Formula: see text] if for any nonnegative integer [Formula: see text], the equation [Formula: see text] has an integer solution [Formula: see text]. In this paper, we prove the universalities of [Formula: see text] ternary sums of generalized polygonal numbers, which was conjectured by Sun.

For an integer x, an integer of the form P 5 pxq " 3x 2´x

Abstract. Let L be a positive definite (non-classic) ternary Z-lattice and let p be a prime such that a 1 2Zp-modular component of Lp is nonzero isotropic and 4¨dL is not divisible by p. For a nonnegative integer m, let G L,p pmq be the genus with discriminant p m¨d L on the quadratic spaceZp-modular component of Tp is nonzero isotropic, and Tq is isometric to pL p m qq for any prime q different from p. Let rpn, M q be the number of representations of an integer n by a Z-lattice M . In this article, we show that if m ď 2 and n is divisible by p only when m " 2, then for any T P G L,p pmq, rpn, T q can be written as a linear summation of rppn, S i q and rpp 3 n, S i q for S i P G L,p pm`1q with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute rpn, T q, for any T P G L,p pmq, by using the number of representations of some integers by lattices in G L,p pm`1q for an arbitrary integer m.