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QUARTERNIONS AND THE FOUR SQUARE THEOREM
, 2008
"... . The Four Square Theorem was proved by Lagrange in 1770: every positive integer is the sum of at most four squares of positive integers, i.e. n = A2+B2+C2+D2, A,B,C,D ∈ Z An interesting proof is presented here based on Hurwitz integers, a subset of quarternions which act like integers in four dim ..."
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. The Four Square Theorem was proved by Lagrange in 1770: every positive integer is the sum of at most four squares of positive integers, i.e. n = A2+B2+C2+D2, A,B,C,D ∈ Z An interesting proof is presented here based on Hurwitz integers, a subset of quarternions which act like integers in four
Jacobi’s two–square theorem and related identities
 The Ramanujan Journal
"... Abstract. We show that Jacobi’s twosquare theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan. Key words: Jacobi’s twosquare theorem, identities of Ramanujan ..."
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Cited by 5 (3 self)
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Abstract. We show that Jacobi’s twosquare theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan. Key words: Jacobi’s twosquare theorem, identities of Ramanujan
FERMAT’S FOUR SQUARES THEOREM
, 2007
"... Fermat proved that there is no arithmetic progression of more than three squares (of rationals). In other words, the pair of diophantine equations a² + c² = 2b² and b² + d² = 2c² has no solution in rationals a, b, c and d. Curiously, the readily accessible literature seems not to contain a straightf ..."
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Fermat proved that there is no arithmetic progression of more than three squares (of rationals). In other words, the pair of diophantine equations a² + c² = 2b² and b² + d² = 2c² has no solution in rationals a, b, c and d. Curiously, the readily accessible literature seems not to contain a
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
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Cited by 9 (3 self)
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simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square
THREESQUARE THEOREM AS AN APPLICATION OF ANDREWS 1 IDENTITY
, 1991
"... The representation of an integer n as a sum of k squares is one of the most beautiful problems in the theory of numbers. Such representations are useful in lattice point problems, crystallography, and certain problems in mechanics [6, pp. 14]. If rk{ri) denotes the number of representations of an i ..."
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of an integer n as a sum of k squares, Jacobi's two and foursquare theorems [9] are:
Nonperiodic Sampling And The Local Three Squares Theorem
"... This paper presents an elementary, realvariable proof of the following theorem: Given fr i g m i=1 with m = d + 1, fix R P m i=1 r i and let Q = [\GammaR; R] d . Then any f 2 L 2 (Q) is completely determined by its averages over cubes of side r i that are completely contained in Q and ha ..."
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Cited by 3 (0 self)
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and have edges parallel to the coordinate axes if and only if r i =r j is irrational for i 6= j. When d = 2 this theorem is known as the local three squares theorem and is an example of a Pompeiutype theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling
An alternate proof of Cohn’sfour squares theorem
, 2002
"... 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 pffiffi representations b ..."
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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 pffiffi representations
Designing an Algorithmic Proof of the TwoSquares Theorem
"... Abstract. We show a new and constructive proof of the twosquares theorem, based on a somewhat unusual, but very effective, way of rewriting the socalled extended Euclid’s algorithm. Rather than simply verifying the result — as it is usually done in the mathematical community — we use Euclid’s algo ..."
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Cited by 1 (0 self)
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Abstract. We show a new and constructive proof of the twosquares theorem, based on a somewhat unusual, but very effective, way of rewriting the socalled extended Euclid’s algorithm. Rather than simply verifying the result — as it is usually done in the mathematical community — we use Euclid’s
A Six Generalized Squares Theorem, with Applications to Polynomial Identity Algebras
, 2000
"... The theories of superalgebras and of P.I. algebras lead to a natural graded2 extension of the integers. For these generalized integers, a ‘‘six generalized squares’ ’ theorem is proved, which can be considered as a graded analogue of2 the classical ‘‘four squares’ ’ theorem for the natural numbers ..."
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The theories of superalgebras and of P.I. algebras lead to a natural graded2 extension of the integers. For these generalized integers, a ‘‘six generalized squares’ ’ theorem is proved, which can be considered as a graded analogue of2 the classical ‘‘four squares’ ’ theorem for the natural
Algebraic Consequences of Jacobi's Two and FourSquare Theorems
 Ismail (eds), Developments in Mathematics
"... . Jacobi's two and foursquare theorems are equivalent to identities in which the generating functions of r 2 (n) and r 4 (n) (respectively the number of representations of n as a sum of 2 and 4 squares) are given both by single infinite products and Lambert series. We present a number of id ..."
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Cited by 2 (1 self)
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. Jacobi's two and foursquare theorems are equivalent to identities in which the generating functions of r 2 (n) and r 4 (n) (respectively the number of representations of n as a sum of 2 and 4 squares) are given both by single infinite products and Lambert series. We present a number
Results 1  10
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