On the representation numbers of ternary quadratic forms and modular forms of weight 3/2

Congruences Modulo 4 for Weight Eta-Products

“…for n ≥ 0 and any prime p > 3. This identity was conjectured by Cooper and Lam [4] and proved by Guo et al [7] (in particular, see [7,Conjecture 1.5]). We note that each of the eta-products g(τ) which occur in parts (i), (ii), (iii) of Theorem 1.3 satisfy ord(g(τ), ∞) = ℓ 24 for ℓ = 11, 19 and 7 respectively.…”

Representations of squares by certain quinary quadratic forms

“…1,5,7 or 11 (mod 24), 1 if n ⌘ 13, 17, 19 or 23 (mod 24)Let m be a positive integer that is relatively prime to 6 and has prime…”

“…Acknowledgment. After submitting this paper for publication, the author learned that all of the cases of Conjecture 1.2 have been proved, by a di↵erent method, by Guo, Peng and Qin [5].…”

On a conjecture of Sun

“…Let n ∈ Z + . For (a, b, c) = (1, 1, 3), (1,1,4), (1,1,6), (1,1,7), (1,1,15), (1,2,2), (1,2,5), (1,3,3), (1,3,9), (1,5,10), (1,6,9), (1,7,7), (1,7,15), (1,9,15), (1,15,15), (1,15,25), (2,3,3) we have 16T (a, b, c; n) = N (a, b, c; 4(8n + a + b + c)) − N (a, b, c; 8n + a + b + c). Conjecture 6.2.…”

“…Let n ∈ Z + . (i) For even n and (a, b, c) = (1,2,15), (1,15,18), (1,15,30), (3,10,45) (ii) For odd n and (a, b, c) = (1,6,7), (1,7,42), (2,3,21), (2,9,15), (3,5,6), (3,5,10), (5,21,35) we have (1,3,5), (1,3,7), (1,3,15), (1,3,21), (1,5,15), (1,7,21), (3,5,9),…”