We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. show Q(5) is the only such field, and among of degrees 3, 4, 5, 7 which have principal codifferent ideal, one Q(ζ7+ζ7−1), over x2+y2+z2+w2+xy+xz+xw universal. Moreover, we prove an upper bound for Pythagoras numbers orders in depends on degree field.

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which locally represented. In this paper, we show that there are at most 147 diagonal ternary forms and prove the odd-regularities of all but six candidates.

Consider the random quadratic form Tn=∑ 1≤u<v≤nauvXuXv, where ((auv))1≤u,v≤n is a {0,1}-valued symmetric matrix with zeros on diagonal, and X1,X2,…,Xn are i.i.d. Ber(pn), pn∈(0,1). In this paper, we prove various characterization theorems about limiting distribution of Tn, in sparse regime, pn→0 such that E(Tn)=O(1). The main result decomposition theorem showing distributional limits Tn sum thr...