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63 lines
1.7 KiB
Plaintext
63 lines
1.7 KiB
Plaintext
NAME
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jacobi - Jacobi symbol function
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SYNOPSIS
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jacobi(x, y)
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TYPES
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x integer
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y integer
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return 1, -1, or 0
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DESCRIPTION
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If y is a positive odd prime and x is an integer not divisible
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by y, jacobi(x,y) returns the Legendre symbol function, usually
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denoted by (x/y) as if x/y were a fraction; this has the value
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1 or -1 according as x is or is not a quadratic residue modulo y.
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x is a quadratic residue modulo y if for some integer u,
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x = u^2 (mod y); if for all integers u, x != u^2 (mod y), x
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is said to be a quadratic nonresidue modulo y.
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If y is a positive odd prime and x is divisible by y, jacobi(x,y)
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returns the value 1. (This differs from the zero value usually
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given in number theory books for (x/y) when x and y
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are not relatively prime.)
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assigned to (x/y) O
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If y is an odd positive integer equal to p_1 * p_2 * ... * p_k,
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where the p_i are primes, not necessarily distinct, the
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jacobi symbol function is given by
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jacobi(x,y) = (x/p_1) * (x/p_2) * ... * (x/p_k).
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where the functions on the right are Legendre symbol functions.
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This is also often usually by (x/y).
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If jacobi(x,y) = -1, then x is a quadratic nonresidue modulo y.
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Equivalently, if x is a quadratic residue modulo y, then
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jacobi(x,y) = 1.
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If jacobi(x,y) = 1 and y is composite, x may be either a quadratic
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residue or a quadratic nonresidue modulo y.
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If y is even or negative, jacobi(x,y) as defined by calc returns
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the value 0.
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EXAMPLE
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> print jacobi(2,3), jacobi(2,5), jacobi(2,15)
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-1 -1 1
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> print jacobi(80,199)
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1
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LIMITS
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none
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LIBRARY
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NUMBER *qjacobi(NUMBER *x, NUMBER *y)
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FLAG zjacobi(ZVALUE z1, ZVALUE z2)
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SEE ALSO
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