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Paley's Theorem


Proved in 1933. If q is an odd prime or q=0 and n is any positive integer, then there is a Hadamard matrix of order

 m=2^e(q^n+1),

where e is any positive integer such that m=0 (mod 4). If m is of this form, the matrix can be constructed with a Paley construction. If m is divisible by 4 but not of the form (1), the Paley class is undefined. However, Hadamard matrices have been shown to exist for all m=0 (mod 4) for m<668.


See also

Hadamard Graph, Hadamard Matrix, Paley Class, Paley Construction, Paley Graph

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Cite this as:

Weisstein, Eric W. "Paley's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PaleysTheorem.html

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