# of squares

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##### 1: 27.17 Other Applications
Apostol and Zuckerman (1951) uses congruences to construct magic squares. …
##### 2: 27.13 Functions
The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. …
###### §27.13(iv) Representation by Squares
For a given integer $k\geq 2$ the function $r_{k}\left(n\right)$ is defined as the number of solutions of the equation … Jacobi (1829) notes that $r_{2}\left(n\right)$ is the coefficient of $x^{n}$ in the square of the theta function $\vartheta\left(x\right)$: … For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.
##### 3: 1.1 Special Notation
 $x,y$ real variables. … the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$. … inverse of the square matrix $\mathbf{A}$ … determinant of the square matrix $\mathbf{A}$ trace of the square matrix $\mathbf{A}$ … adjoint of the square matrix $\mathbf{A}$ …
##### 4: 8.23 Statistical Applications
###### §8.23 Statistical Applications
Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …
##### 5: 19.38 Approximations
Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …
##### 6: 27.22 Software
• Maple. isprime combines a strong pseudoprime test and a Lucas pseudoprime test. ifactor uses cfrac27.19) after exhausting trial division. Brent–Pollard rho, Square Forms Factorization, and ecm are available also; see §27.19.

• ##### 7: 34.14 Tables
Tables of exact values of the squares of the $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols on pp. …
##### 8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$
For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$, …The space $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ becomes a separable Hilbert space with inner product … Eigenfunctions corresponding to the continuous spectrum are non-$L^{2}$ functions. … The well must be deep and broad enough to allow existence of such $L^{2}$ discrete states. …
##### 10: 3.11 Approximation Techniques
###### §3.11(v) Least Squares Approximations
For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …