By means of q-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x 2 + y 2 for some x, y ∈ Z with x ̸ ≡ y (mod 2) or x = y> 0. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.

this paper to indicate how continued fractions are relevant to

Abstract. The problem of finding all quadratic forms over Z that represent each positive integer received significant attention in a paper of Ramanujan in 1917. Exactly fifty four quaternary quadratic forms of this type without cross product terms were shown to represent all positive integers. The classical case of the quadratic form that is just the sum of four squares received an alternate proof by Hurwitz using a special ring of quaternions. Here we prove that seven other quaternary quadratic forms can be shown to represent all positive integers by investigation of the corresponding quaternion rings. 1.

Dedicated to the memory of Arnold E. Ross Let q(x) = q(x1, x2, x3) be a positive definite ternary quadratic form with integral coefficients. In 1946 Ross and Pall [RP] conjectured that every sufficiently large square-free integer that is represented by q modulo N for all N is in fact

Abstract. This paper is a slightly enlarged version of a series of lectures on the Linnik problems

In this paper we show that, under the assumption that all the zeros of the L-functions under consideration are either real or lie on the critical line, one may considerably improve on the known results on Landau-Siegel zeros. 1.

Abstract. The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper we develop results analogous to Hurwitz’s above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form. 1.

Abstract. By means of q-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x 2 + y 2 for some integers x> 0 and y � 0 with x̸ ≡ y (mod 2) or x = y. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers. 1.

Abstract. In this paper we investigate mixed sums of squares and triangular numbers for the first time. We prove that any natural number n can be written as x 2 + y 2 + Tz with x, y, z ∈ Z and Tz = z(z + 1)/2. Also, we can express n in any of the following forms:

Abstract. In this paper we investigate mixed sums of squares and triangular numbers. By means of q-series, we prove that any natural number n can be written as x 2 + (2y) 2 + Tz with x, y, z ∈ Z and Tz = z(z + 1)/2, this is stronger than a conjecture of Chen. Also, we can express n in any of the following forms: