magic 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 $k\ge 2$ 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: Bibliography

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On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.

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External Links:

ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

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4: 1.1 Special Notation

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$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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${𝐀}^{-1}$	inverse of the square matrix $𝐀$
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$det(𝐀)$	determinant of the square matrix $𝐀$
$\mathrm{tr}(𝐀)$	trace of the square matrix $𝐀$
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${𝐀}^{\ast }$	adjoint of the square matrix $𝐀$
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5: 8.23 Statistical Applications

§8.23 Statistical Applications

… ►Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …

6: 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). …

7: 27.22 Software

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Maple. isprime combines a strong pseudoprime test and a Lucas pseudoprime test. ifactor uses cfrac (§27.19) after exhausting trial division. Brent–Pollard rho, Square Forms Factorization, and ecm are available also; see §27.19.

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8: 34.14 Tables

… ►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. …

9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$, …The space $L^2(X,d\alpha )$ 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: 1.2 Elementary Algebra

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§1.2(vi) Square Matrices

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Special Forms of Square Matrices

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Norms of Square Matrices

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Non-Defective Square Matrices

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