Four Squares
نویسندگان
چکیده
منابع مشابه
Arithmetic Progressions of Four Squares
Suppose a, b, c, and d are rational numbers such that a2, b2, c2, and d2 form an arithmetic progression: the differences b2−a2, c2−b2, and d2−c2 are equal. One possibility is that the arithmetic progression is constant: a2, a2, a2, a2. Are there arithmetic progressions of four rational squares which are not constant? This question was first raised by Fermat in 1640. There are no such progressio...
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This document gives the formal proofs of the following results about the sums of two and four squares: 1. Any prime number p ≡ 1 mod 4 can be written as the sum of two squares. 2. (Lagrange) Any natural number can be written as the sum of four
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In recent work [1,2], we studied p4 (n), the number of partitions of n into four squares. We found the generating function for p4 (n), and showed that the numbers p4 (n) possess some (rare) arithmetic properties. In this paper we make a study of what we will call p4e(n) and p4o(n), the number of partitions of n into four distinct even (respectively odd) squares. We shall also have reason to con...
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We prove several results dealing with various counting functions for partitions of an integer into four squares of equal parity. Some are easy consequences of earlier work, but two are new and surprising. That is, we show that the number of partitions of 72n+60 into four odd squares (distinct or not) is even.
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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 ofQð ffiffiffi 5 p Þ was demonstrated by Götzky and the cases of Qð ffiffiffi 2 p Þ and Qð ffiffiffi 3 p Þ were demonstrated by Cohn. In the latter situation, only those integers with even coefficient on the radical term could possibl...
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ژورنال
عنوان ژورنال: Writing From the World: Selections from the International Writing Program 1977-1983
سال: 1984
ISSN: 0021-065X,2330-0361
DOI: 10.17077/0021-065x.3064