NAME
    jacobi - Jacobi symbol function

SYNOPSIS
    jacobi(x, y)

TYPES
    x		integer
    y		integer

    return	1, -1, or 0

DESCRIPTION
    The jacobi builtin function of calc returns the value of the
    Jacobi symbol.

    If y is an odd prime, then the Legendre symbol (x/y) returns:

	(x/y) == 0	x is divisible by y (0 == x % y)
	(x/y) == 1	if x is a quadratic residue modulo y
	(x/y) == -1	if x is not a quadratic residue modulo y

    Legendre symbol often denoted as (x/y) as if x/y were a fraction.

    If there exists an integer u such that:

	x == u^2 modulo y		(i.e., x == u^2 % y)

    then x is a quadratic residue modulo y.  However, if for all integers u:

	x != u^2 modulo y		(i.e., x != u^2 % y)

    x is said to be a quadratic non-residue modulo y.

    When y is prime, Jacobi symbol is the same as the Legendre symbol.
    That is, Jacobi symbol satisfies the same rules as the Legendre symbol.
    Therefore, when y is an odd prime:

	jacobi(x,y) == 0	x is divisible by y (x == 0 % y)
	jacobi(x,y) == 1	if x is a quadratic residue modulo y
	jacobi(x,y) == -1	if x is a quadratic non-residue modulo y

    When y is even or y <= 0:

	jacobi(x,y) == 0

    When y is not prime, we may express y as the product of primes:

	y == p0^a0 * p1^a1 * .. * pn^an

    When y is odd and NOT prime, then:

	jacobi(x,y) == (x/p0)^a0 * (x/p1)^a1 * .. * (x/pn)^an

    where (x/y) is Legendre symbol.

    If x is a quadratic residue modulo y, then jacobi(x,y) == 1.

    If jacobi(x,y) == 1 and y is composite (y is not prime), x may
    be either a quadratic residue or a quadratic non-residue modulo y.

    If y is even or negative, jacobi(x,y) as defined by calc, returns 0.

    Jacobi symbol details
    ---------------------

    In the rest of this section we present for the curious, some
    rules related to the jacobi symbol.  Most of these rules are
    used internally by calc to evaluate the jacobi symbol.

    In this section, a and b are positive odd integers.
    In this section, x and y are any integers.

    We have these identities:

	jacobi(x,y) == 0			if x <= 0
	jacobi(x,y) == 0			if y <= 0
	jacobi(x,y) == 0			if y is even

	jacobi(x,b) == 0			if gcd(x,b) > 1

	jacobi(0,b) == 0
	jacobi(1,b) == 1

	jacobi(x,b) == jacobi(y,b)		if x == y % b

	jacobi(a,b) == jacobi(2*a,b)		if b == 1 % 8 OR b == 7 % 8
	jacobi(a,b) == -jacobi(2*a,b)		if b != 1 & 8 AND b != 7 % 8

	jacobi(a,b) == jacobi(b,a)		if a == 3 % 4 AND b == 3 % 4
	jacobi(a,b) == -jacobi(b,a)		if a != 3 % 4 OR b != 3 % 4

	jacobi(-1,b) == 1			if b == 1 % 4
	jacobi(-1,b) == -1			if b == 3 % 4

	jacobi(x,b) * jacobi(y,b) == jacobi(x*y,b)
	jacobi(x,a) * jacobi(x,b) == jacobi(x,a*b)

    Technically the Jacobi symbol jacobi(x,y) is not defined when
    x == 0, x < 0, y is even, y == 0, and when y < 0.  So by convention:

	jacobi(x,y) == 0			if x <= 0
	jacobi(x,y) == 0			if y <= 0
	jacobi(x,y) == 0			if y == 0 % 2

    It is also worth noting that:

	jacobi(x,y) == 0			if gcd(y,x) != 1
	jacobi(y,y) == 0			if y > 1

    Based on the generalization of the quadratic reciprocity theorem,
    when a and b are odd, b >= 3, and gcd(a,b) == 1:

	jacobi(a,b) == jacobi(b,a) * ((-1) ^ ((a-1) * (b-1) / 4))

    Therefore, when a and b are odd, b >= 3, and gcd(a,b) == 1:

	jacobi(a,b) == jacobi(b,a)		if a == 1 % 4 OR b == 1 % 4
	jacobi(a,b) == -jacobi(b,a)		if a == 3 % 4 AND b == 3 % 4

    Jacobi symbol satisfies the same rules as the Legendre symbol such as:

	jacobi(x,y0*y1) == jacobi(x,y0) * jacobi(x,y1)
	jacobi(x0*x1,y) == jacobi(x0,y) * jacobi(x1,y)

    When b is odd:

	jacobi(x^2,b) == 1				if gcd(x,b) == 1
	jacobi(x,b^2) == 1				if gcd(x,b) == 1

	jacobi(x0,b) == jacobi(x1,b)			if x0 == x1 % b

	jacobi(-1,b) == (-1) ^ ((b-1) / 2) == 1		if b == 1 % 4
	jacobi(-1,b) == (-1) ^ ((b-1) / 2) == -1	if b == 3 % 4

	jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == 1	if b == 1 % 8 OR
							   b == 7 % 8
	jacobi(2,b) == (-1) ^ ((b^2 - 1) / 8) == -1	if b == 3 % 8 OR
							   b == 5 % 8

	jacobi(-3,b) == 1				if b == 1 % 6
	jacobi(-3,b) == -1				if b == 5 % 6

	jacobi(5,b) == 1				if b == 1 % 10 OR
							   b == 9 % 10
	jacobi(5,b) == -1				if b == 3 % 10 OR
							   b == 7 % 10

    If a and b are both odd primes, OR
    if gcd(a,b) = 1 and, a > 0, and a is odd:

	jacobi(a,b) == jacobi(b,a) * ((-1) ^ (((a-1)/2) * ((b-1)/2)))
	jacobi(a,b) * jacobi(b,a) == ((-1) ^ (((a-1)/2) * ((b-1)/2)))

    If b is an odd prime:

	jacobi(a,b) == (a ^ ((b-1)/2)) % b

    Combining all of the above, computing jacobi(x,b) may be expressed
    in the following algorithm when b is odd and 0 < x < b:

	j = 1;
	while (x != 0) {
	    while (iseven(x)) {
		x = x / 2;
		t = b % 8;
		if (t == 3 || t == 5) {
		    j = -j;
		}
	    }
	    swap(x,b);
	    if (((x % 4) == 3) && ((b % 4) == 3)) {
		j = -j;
	    }
	    x = x % b;
	}

	if (b == 1) {
	    return j;
	} else {
	    return 0;
	}

    The above calc pseudo code is provided to help illustrate the
    properties of the jacobi symbol only.  The use of calc's jacobi
    builtin is recommended for both speed as well as to handle
    special cases.

    For more information on the Legendre symbol and the Jacobi symbol:

	http://mathworld.wolfram.com/LegendreSymbol.html
	http://mathworld.wolfram.com/JacobiSymbol.html
	http://primes.utm.edu/glossary/xpage/JacobiSymbol.html

EXAMPLE
    ; print jacobi(2,3), jacobi(2,5), jacobi(2,15)
    -1 -1 1

    ; print jacobi(80,199)
    1

LIMITS
    none

LINK LIBRARY
    NUMBER *qjacobi(NUMBER *x, NUMBER *y)
    FLAG zjacobi(ZVALUE z1, ZVALUE z2)

SEE ALSO

## Copyright (C) 1999,2017,2021  Landon Curt Noll
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## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
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## Under source code control:	1995/12/18 12:34:57
## File existed as early as:	1995
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## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
