# 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: 8.23 Statistical Applications
###### §8.23 Statistical Applications
Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …
##### 4: 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). …
##### 5: 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.

• ##### 6: 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. …
##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$
Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval 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 $\,\mathrm{d}\alpha$ ,Functions $f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)$ for which $\left\langle f-g,f-g\right\rangle=0$ are identified with each other. The space $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ becomes a separable Hilbert space with inner productAssume that $\left\{\phi_{n}\right\}_{n=0}^{\infty}$ is an orthonormal basis of $L^{2}\left(X\right)$ . The formulas in §1.18(i) are then:for $f(x)\in L^{2}$ and piece-wise continuous, with convergence as discussed in §1.18(ii).
##### 8: 18.36 Miscellaneous Polynomials
These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. …
##### 9: 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). …
##### 10: 19.31 Probability Distributions
$R_{G}\left(x,y,z\right)$ and $R_{F}\left(x,y,z\right)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$, of the square root or reciprocal square root of a quadratic form. …