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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… (More)

While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in Qð ffiffi ffi 5 p Þ was resolved by… (More)

- Jesse I. Deutsch
- 2008

In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville. Another quadratic form over the field of discriminant 8 is shown universal by a different elementary… (More)

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… (More)

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontriv-ial isotropy in the fundamental domains under the action of the modular group for Q(√ 5), Q(√ 2), and Q(√ 3) have been determined previously by Gundlach. In finding these points, use was made of the exact size of the… (More)

The totally positive algebraic integers of certain number fields have been shown to be the sums of four squares of integers from their respective fields. The case of Qð ffiffi ffi 5 p Þ was demonstrated by Go¨tzky and the cases of Qð ffiffi ffi 2 p Þ and Qð ffiffi ffi 3 p Þ were demonstrated by Cohn. In the latter situation, only those integers with even… (More)

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