Cooper and Lam's conjecture for generalized Bell ternary quadratic forms

Quadratic forms with a strong regularity property on the representations of squares

“…Furthermore, they provided a list of 64 diagonal ternary quadratic forms and conjectured that those ternary forms in the list also satisfy similar equations. Some parts of their conjecture were proved by Guo, Peng, and Qin [6], Hürlimann [11], and finally, all of the remaining cases were proved by the authors [14].…”

On a conjecture of Sun

“…Lemma 2.2. The following identities hold: ϕ(q) = ϕ(q 4 ) + 2qψ(q 8 ), (7) ϕ(q) 2 = ϕ(q 2 ) 2 + 4qψ(q 4 ) 2 , (8) ψ(q) 2 = ψ(q 2 )ϕ(q 4 ) + 2qψ(q 2 )ψ(q 8 ), (9) ψ(q)ψ(q 3 ) = ψ(q 4 )ϕ(q 6 ) + qϕ(q 2 )ψ(q 12 ), (10) ϕ(q)ϕ(q 3 ) = ϕ(q 4 )ϕ(q 12 ) + 2qψ(q 2 )ψ(q 6 ) + 4q 4 ψ(q 8 )ψ(q 24 ), (11) ψ(q)ψ(q 7 ) = ψ(q 8 )ϕ(q 28 ) + qψ(q 2 )ψ(q 14 ) + q 6 ϕ(q 4 )ψ(q 56 ), (12) ψ(q)ψ(q 15 ) = ψ(q 6 )ψ(q 10 ) + qϕ(q 20 )ψ(q 24 ) + q 3 ϕ(q 12 )ψ(q 40 ), (13) ψ(q 3 )ψ(q 5 ) = ψ(q 8 )ϕ(q 60 ) + q 3 ψ(q 2 )ψ(q 30 ) + q 14 ϕ(q 4 )ψ(q 120 ). (14) The proof of this lemma is omitted; please look at [17] for the same.…”

“…Recently, Xia and Yan [17] proved Conjecture 6.1 for (1, 1, 7), (1,1,15), (1,7,7), (1,7,15), (1,9,15), (1,15,15), (1,15,25); Conjecture 6.2 (i) for (1,2,15), (1,15,18), (1,15,30); and Conjecture 6.3. For more related works, please look at [2,4,5,7,8,9,10,11,12,13,14,15,18,19,20,21,22].…”

Spinor representations of positive definite ternary quadratic forms

“…As noted in the introduction, it is well known that there are exactly 12 generalized Bell ternary Z-lattices having class number 1. For each of these 12 lattices, it is proved in [1] and [5] that the number of representations of an integer k can be written as a constant multiple of the number of representations of an integer, which is not necessarily same to k, by a sum of three squares. In this section, we prove similar results in the case when the spinor genus of a generalized Bell ternary lattice contains only one class.…”