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Some folks might think: “you still use RCS”?!? And we will say, hey, at least we switched from SCCS to RCS back in … I think it was around 1994 ... at least we are keeping up! :-) :-) :-) Logs say that SCCS version 18 became RCS version 19 on 1994 March 18. RCS served us well. But now it is time to move on. And so we are switching to git. Calc releases produce a lot of file changes. In the 125 releases of calc since 1996, when I started managing calc releases, there have been 15473 file mods!
232 lines
6.4 KiB
Plaintext
232 lines
6.4 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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The jacobi builtin function of calc returns the value of the
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Jacobi symbol.
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If y is an odd prime, then the Legendre symbol (x/y) returns:
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(x/y) == 0 x is divisible by y (0 == x % y)
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(x/y) == 1 if x is a quadratic residue modulo y
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(x/y) == -1 if x is not a quadratic residue modulo y
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Legendre symbol often denoted as (x/y) as if x/y were a fraction.
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If there exists an integer u such that:
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x == u^2 modulo y (i.e., x == u^2 % y)
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then x is a quadratic residue modulo y. However, if for all integers u:
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x != u^2 modulo y (i.e., x != u^2 % y)
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x is said to be a quadratic non-residue modulo y.
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When y is prime, Jacobi symbol is the same as the Legendre symbol.
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That is, Jacobi symbol satisfies the same rules as the Legendre symbol.
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Therefore, when y is an odd prime:
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jacobi(x,y) == 0 x is divisible by y (x == 0 % y)
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jacobi(x,y) == 1 if x is a quadratic residue modulo y
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jacobi(x,y) == -1 if x is a quadratic non-residue modulo y
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When y is even or y <= 0:
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jacobi(x,y) == 0
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When y is not prime, we may express y as the product of primes:
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y == p0^a0 * p1^a1 * .. * pn^an
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When y is odd and NOT prime, then:
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jacobi(x,y) == (x/p0)^a0 * (x/p1)^a1 * .. * (x/pn)^an
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where (x/y) is Legendre symbol.
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If x is a quadratic residue modulo y, then jacobi(x,y) == 1.
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If jacobi(x,y) == 1 and y is composite (y is not prime), x may
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be either a quadratic residue or a quadratic non-residue modulo y.
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If y is even or negative, jacobi(x,y) as defined by calc, returns 0.
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Jacobi symbol details
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---------------------
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In the rest of this section we present for the curious, some
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rules related to the jacobi symbol. Most of these rules are
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used internally by calc to evaluate the jacobi symbol.
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In this section, a and b are positive odd integers.
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In this section, x and y are any integers.
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We have these identities:
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jacobi(x,y) == 0 if x <= 0
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jacobi(x,y) == 0 if y <= 0
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jacobi(x,y) == 0 if y is even
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jacobi(x,b) == 0 if gcd(x,b) > 1
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jacobi(0,b) == 0
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jacobi(1,b) == 1
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jacobi(x,b) == jacobi(y,b) if x == y % b
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jacobi(a,b) == jacobi(2*a,b) if b == 1 % 8 OR b == 7 % 8
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jacobi(a,b) == -jacobi(2*a,b) if b != 1 & 8 AND b != 7 % 8
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jacobi(a,b) == jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
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jacobi(a,b) == -jacobi(b,a) if a != 3 % 4 OR b != 3 % 4
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jacobi(-1,b) == 1 if b == 1 % 4
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jacobi(-1,b) == -1 if b == 3 % 4
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jacobi(x,b) * jacobi(y,b) == jacobi(x*y,b)
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jacobi(x,a) * jacobi(x,b) == jacobi(x,a*b)
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Technically the Jacobi symbol jacobi(x,y) is not defined when
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x == 0, x < 0, y is even, y == 0, and when y < 0. So by convention:
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jacobi(x,y) == 0 if x <= 0
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jacobi(x,y) == 0 if y <= 0
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jacobi(x,y) == 0 if y == 0 % 2
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It is also worth noting that:
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jacobi(x,y) == 0 if gcd(y,x) != 1
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jacobi(y,y) == 0 if y > 1
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Based on the generalization of the quadratic reciprocity theorem,
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when a and b are odd, b >= 3, and gcd(a,b) == 1:
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jacobi(a,b) == jacobi(b,a) * ((-1) ^ ((a-1) * (b-1) / 4))
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Therefore, when a and b are odd, b >= 3, and gcd(a,b) == 1:
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jacobi(a,b) == jacobi(b,a) if a == 1 % 4 OR b == 1 % 4
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jacobi(a,b) == -jacobi(b,a) if a == 3 % 4 AND b == 3 % 4
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Jacobi symbol satisfies the same rules as the Legendre symbol such as:
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jacobi(x,y0*y1) == jacobi(x,y0) * jacobi(x,y1)
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jacobi(x0*x1,y) == jacobi(x0,y) * jacobi(x1,y)
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When b is odd:
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jacobi(x^2,b) == 1 if gcd(x,b) == 1
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jacobi(x,b^2) == 1 if gcd(x,b) == 1
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jacobi(x0,b) == jacobi(x1,b) if x0 == x1 % b
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jacobi(-1,b) == (-1) ^ ((b-1) / 2) == 1 if b == 1 % 4
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jacobi(-1,b) == (-1) ^ ((b-1) / 2) == -1 if b == 3 % 4
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jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == 1 if b == 1 % 8 OR
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b == 7 % 8
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jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == -1 if b == 3 % 8 OR
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b == 5 % 8
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jacobi(-3,b) == 1 if b == 1 % 6
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jacobi(-3,b) == -1 if b == 5 % 6
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jacobi(5,b) == 1 if b == 1 % 10 OR
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b == 9 % 10
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jacobi(5,b) == -1 if b == 3 % 10 OR
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b == 7 % 10
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If a and b are both odd primes, OR
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if gcd(a,b) = 1 and, a > 0, and a is odd:
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jacobi(a,b) == jacobi(b,a) * ((-1) ^ (((a-1)/2) * ((b-1)/2)))
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jacobi(a,b) * jacobi(b,a) == ((-1) ^ (((a-1)/2) * ((b-1)/2)))
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If b is an odd prime:
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jacobi(a,b) == (a ^ ((b-1)/2)) % b
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Combining all of the above, computing jacobi(x,b) may be expressed
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in the following algorithm when b is odd and 0 < x < b:
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j = 1;
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while (x != 0) {
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while (iseven(x)) {
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x = x / 2;
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t = b % 8;
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if (t == 3 || t == 5) {
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j = -j;
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}
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}
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swap(x,b);
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if (((x % 4) == 3) && ((b % 4) == 3)) {
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j = -j;
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}
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x = x % b;
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}
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if (b == 1) {
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return j;
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} else {
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return 0;
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}
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The above calc pseudo code is provided to help illustrate the
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properties of the jacobi symbol only. The use of calc's jacobi
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builtin is recommended for both speed as well as to handle
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specical cases.
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For more information on the Legendre symbol and the Jacobi symbol:
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http://mathworld.wolfram.com/LegendreSymbol.html
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http://mathworld.wolfram.com/JacobiSymbol.html
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http://primes.utm.edu/glossary/xpage/JacobiSymbol.html
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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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LINK 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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## Copyright (C) 1999,2017 Landon Curt Noll
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##
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## Calc is open software; you can redistribute it and/or modify it under
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## the terms of the version 2.1 of the GNU Lesser General Public License
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## as published by the Free Software Foundation.
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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## distributed with calc under the filename COPYING-LGPL. You should have
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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##
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## Under source code control: 1995/12/18 12:34:57
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## File existed as early as: 1995
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##
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## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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