Legendre’s Theorem

Emil Grosswald
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Emil Grosswald: Temple University, College of Liberal Arts

Chapter Chapter 5 in Representations of Integers as Sums of Squares, 1985, pp 66-71 from Springer

Abstract: Abstract In this chapter we consider a diagonal form more general than a simple sum of squares. We shall be concerned with the ternary quadratic forms Q(x,y,z) = ax2 + by2 + cz2. If a, b, c are positive integers, then strictly speaking this form is also a sum of squares, because it can be written as $$ \underbrace{{{x^2} + {x^2} + \cdots + {x^2}}}_{{a\;{\text{time}}}} + \underbrace{{{y^2} + {y^2} + \cdots + {y^2}}}_{{b\;{\text{time}}}} + \underbrace{{{z^2} + {z^2} + \cdots + {z^2}}}_{{c\;{\text{time}}}} $$ but the number of squares varies with the values of the coefficients a, b, c. In fact, however, Q is an arbitrary diagonal form, with a, b, c integers, but not necessarily positive.

Date: 1985
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DOI: 10.1007/978-1-4613-8566-0_6

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