The number of representations of by squares, allowing zeros and distinguishing signs and order, is denoted . The special case corresponding to two squares is often denoted simply (e.g., Hardy and Wright 1979, p. 241; Shanks 1993, p. 162).
For example, consider the number of ways of representing 5 as the sum of two squares:
so . Similarly,
The Wolfram Language function SquaresR[k, n] gives . In contrast, the function PowersRepresentations[n, k, 2] gives a list of unordered unsigned representations of as a list of squares, e.g., giving the as the only "unique" representation of 5.
The function is intimately connected with the Leibniz series and with Gauss's circle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39). It is also given by the inverse Möbius transform of the sequence and (Sloane and Plouffe 1995, p. 22). The average order of is , but the normal order is 0 (Hardy 1999, p. 55).
Jacobi gave analytic expressions for for the cases , 4, 6, and 8 (Jacobi 1829; Hardy and Wright 1979, p. 316; Hardy 1999, p. 132). The cases , 4, and 6 were found by equating coefficients of the Jacobi theta functions , , and . The solutions for and 12 were found by Liouville (1864, 1866) and Eisenstein (Hardy and Wright 1979, p. 316), and Glaisher (1907) gives a table of for up to . However, the formulas for and contained functions defined only as the coefficients of modular functions, but not arithmetically (Hardy and Wright 1979, p. 316). Ramanujan (2000) extended Glaisher's table up to . Boulyguine (1915) found a general formula for in which every function has an arithmetic definition (Hardy and Wright 1979, p. 316; Dickson 2005, p. 317).
was found as a finite sum involving quadratic reciprocity symbols by Dirichlet. and were found by Eisenstein, Smith, and Minkowski. Mordell, Hardy, and Ramanujan have developed a method applicable to representations by an odd number of squares (Hardy 1920; Mordell 1920, 1923; Estermann 1937; Hardy 1999).
To find in how many ways a positive integer can be expressed as a sum of squares ignoring order and signs, factor it as
where the s are primes of the form and the s are primes of the form . If does not have such a representation with integer because one or more of the powers of is odd, then there are no representations. Otherwise, define
The number of representations of as the sum of two squares ignoring order and signs is then given by
(Beiler 1966, pp. 140-142).
Similarly, for is given by
A positive integer can be represented as the sum of two squares iff each of its prime factors of the form occurs as an even power, as first established by Euler in 1738. In Lagrange's four-square theorem, Lagrange proved that every positive integer can be written as the sum of at most four squares, although four may be reduced to three except for numbers of the form .
Diophantus first studied a problem equivalent to finding three squares whose sum is , and stated that for this problem, must not be of the form , which is however an insufficient condition (Dickson 2005, p. 259). In 1621, Bachet subsequently excluded and . Finally, Fermat (ca. 1636) remarked that Bachet's condition failed to exclude , 149, etc., and gave the correct sufficient condition that must not be of the form , so not of the form , or equivalently .
In 1636, Fermat stated that no integer of the form is the sum of three rational squares, and in 1638, Descartes proved this for integer squares. In 1658, Fermat subsequently asserted (but did not prove) that , where is any prime of the form (i.e., any prime of the form ) is the sum of three squares. In 1775, Lagrange made some progress on Fermat's assertion, but could not completely prove it. In 1785, Legendre remarked that Fermat's assertion is true for all odd numbers (not just primes), and then gave an incomplete proof that either every number or its double is a sum of three squares.
Beguelin (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). Then, in Legendre's 1798 Théorie des nombres, Legendre proved that every positive integer not of the form or is a sum of three squares having no common factor (Nagell 1951, p. 194; Wells 1986, pp. 48 and 56; Hardy 1999, p. 12; Savin 2000).
is 0 whenever has a prime divisor of the form to an odd power; it doubles upon reaching a new prime of the form . The first few values are 1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, ... (OEIS A004018). A Lambert series was given by
(Hardy and Wright 1979, p. 258). The generating function for is given by
where is a Jacobi elliptic function and is a q-Pochhammer symbol.
It is given explicitly by
where is the number of divisors of of the form (Hilbert and Cohn-Vossen 1999, pp. 37-38; Hardy 1999, p. 12).
obeys the unexpected identities
(Hardy 1999, p. 82).
The first few values of the summatory function (e.g., Hardy and Wright 1979, p. 270) defined by
are 0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36, ... (OEIS A014198),where the modified function defined by Shanks (1993) is
Explicit values of for several powers of 10 are given in the following table (Mitchell 1966; Shanks 1993, pp. 165 and 234).
Asymptotic results include
where is a constant known as the Sierpiński constant. The left plot above shows
with illustrated by curved envelope, and the right plot shows
with the value of indicated as the solid horizontal line.
The number of solutions of
for a given without restriction on the signs or relative sizes of , , and is given by . Gauss proved that if is squarefree and , then
(Arno 1992), where is the class number of .
The generating function for is given by
and in general,
Identities for , and are given by
(Jacobi 1829, §40-42; Smith 1965; Hardy and Wright 1979, p. 314).
This equation and that for were given by Liouville (1864, 1866).
is a so-called singular series, and is the tau function.
Similar expressions exist for larger even , but they quickly become extremely complicated and can be written simply only in terms of expansions of modular functions.