![]() Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. "The Representation of Numbers as Sums of Squares." Ch. 9 "On the Representation of a Number as the Sum of Any Number of Squares, and "On the Expression of a Number as the Sum of Two Squares." Quart. "On the Numbers of a Representation of a Number as a Sum of Squares, where "New Representations of Ramanujan's Tau Function." Proc.Īmer. The Representation of a Number as a Sum of Squares." Acta Arith. "Zur Theorie der quadratischen Zerfällung "Note sur la représentationĭ'un nombre par la somme de cinq carrés." J. Of the Theory of Numbers, Vol. 2: Diophantine Analysis. "Sur la représentationĭ'un nombre entier par une somme de carrés." Comptes Rendus Hebdomadairesĭe Séances de l'Académie des Sciences 161, 28-30, 1915. In the Theory of Numbers: The Queen of Mathematics Entertains. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. Squares having no common factor (Nagell 1951, p. 194 Wells 1986, pp. 48ĭivisor of the form to an odd power it doubles upon reaching a new prime of the form. Then, in Legendre's 1798 Théorie des nombres, Legendre proved thatĮvery positive integer not of the form or is a sum of three Numbers (not just primes), and then gave an incomplete proof that either every numberīeguelin (1774) had concluded that every integer congruent to 1, 2, 3, 5 or 6 (mod 8) is a sum of three squares, but without adequate proof (Dickson 2005, p. 15). In 1785, Legendre remarked that Fermat's assertion is true for all odd Inġ775, Lagrange made some progress on Fermat's assertion, but could not completely The form (i.e., any prime of the form ) is the sum of three squares. In 1658, Fermat subsequently asserted (but did not prove) that, where is any prime of Of three rational squares, and in 1638, Descartes proved this for integer squares. In 1636, Fermat stated that no integer of the form is the sum ![]() 1636) remarked that Bachet's condition failed to exclude, 149, etc.,Īnd gave the correct sufficient condition that must not be of Must not be of the form, which is howeverĪn insufficient condition (Dickson 2005, p. 259). Integer can be written as the sum of at most four squares, although four may be reduced to three exceptĭiophantus first studied a problem equivalent to finding three squares whose sum is, and stated that for this problem, In Lagrange'sįour-square theorem, Lagrange proved that every positive To find in how many ways a positive integer can be expressedĪs a sum of squares ignoring order and signs,Ī positive integer can be represented as the sum of two squares iff each of its primeĮven power, as first established by Euler in 1738. Mordell, Hardy, and Ramanujan have developed a method applicable to representationsīy an odd number of squares (Hardy 1920 Mordell 1920, 1923 Estermann 1937 Hardy and were found by Eisenstein, Smith, and Minkowski. Quadratic reciprocity symbols by Dirichlet. Boulyguine (1915) found a generalĪn arithmetic definition (Hardy and Wright 1979, p. 316 Dickson 2005, p. 317). Ramanujan (2000) extended Glaisher's table Solutions for and 12 were found by Liouville (1864,ġ866) and Eisenstein (Hardy and Wright 1979, p. 316), and Glaisher (1907) givesįunctions defined only as the coefficients of modular functions, but not arithmetically The cases, 4, and 6 wereįound by equating coefficients of the Jacobi Jacobi gave analytic expressions for for the cases It is also givenīy the inverse Möbius transform of the sequence and (Sloane and Plouffe 1995, p. 22). The function is intimately connected with theĬircle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39). K, 2] gives a list of unordered unsigned representations of as a list of squares, e.g., giving the as the In contrast, the function PowersRepresentations[ n, The Wolfram Language function SquaresR[ k,
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