Documentation |
On this page… |
---|
If a, b, and m are integers, and (a - b)/m is also an integer, then the numbers a and b are congruent modulo m. The remainder of the division a/m is equal to the remainder of the division of b/m. For example, 11 ≡ 5(mod 3):
5 mod 3 = 11 mod 3
For known integers a and m, all integers b, such that a ≡ b(mod m), form a residue class. Thus, the numbers 5 and 11 belong to the same residue class modulo 3. The numbers 5 + 3n, where n is an integer, also belong to this residue class.
Suppose, you want to solve an equation ax ≡ b(mod m), where a, b, and m are integers and x is an unknown integer. Such equations are called linear congruence equations. To solve a linear congruence equation, use the numlib::lincongruence function. This function returns only the solutions x < m. For example, solve the linear congruence equation 56x ≡ 77(mod 49):
numlib::lincongruence(56, 77, 49)
A linear congruence equation ax ≡ b(mod m) has at least one solution if and only if the parameters a, b, and m satisfy the following condition: b ≡ 0(mod gcd(a, m)). If the parameters of a linear congruence equation do not satisfy this condition, the equation does not have a solution. In this case, numlib::lincongruence returns FAIL:
numlib::lincongruence(56, 77, 48)
The Chinese remainder theorem states that if the integers m_{i}(i = 1, ..., n) are pairwise coprime, the system of n linear congruences x ≡ a_{i}(mod m_{i}) has a solution. The numbers m_{i}(i = 1, ..., n) are pairwise coprime if the greatest common divisor of any pair of numbers m_{i}, m_{j} (i ≠ j) is 1. The solution is unique up to multiples of the least common multiple (ilcm) of m_{1}, m_{2}, ..., m_{n}. To solve a system of linear congruence equations, use the numlib::ichrem function:
numlib::ichrem([3, 1, 10], [6, 5, 13])
The Chinese remainder theorem does not state that the system of linear congruences is solvable only if numbers m_{1},... m_{n} are pairwise coprime. If these numbers are not pairwise coprime, the system still can have a solution. Even if the numbers are not pairwise coprime, the solution is still unique up to multiples of the least common multiple (ilcm) of m_{1}, m_{2}, ..., m_{n}:
numlib::ichrem([5, 7, 9, 6], [10, 11, 12, 13])
If the numbers are not pairwise coprime, a system of linear congruences does not always have a solution. For unsolvable systems, numlib::ichrem returns FAIL:
numlib::ichrem([5, 1, 9, 6], [10, 15, 12, 13])
To compute modular square roots x < m of the equation x^{2}≡ a(mod m), use the numlib::msqrts function. Here the integers a and m must be coprime. For example, solve the congruence equation x^{2}≡ 13(mod 17):
numlib::msqrts(13, 17)
If the congruence does not have any solutions, numlib::msqrts returns an empty set:
numlib::msqrts(10, 17)
If a and m are not coprime, numlib::msqrts errors:
numlib::msqrts(17, 17)
Error: Arguments must be relative prime. [numlib::msqrts]
If numlib::msqrts cannot solve a congruence, try using the numlib::mroots function. For more information, see General Solver for Congruences.
The Legendre symbol determines the solvability of the congruence x^{2}≡ a(mod m), where m is a prime. You can compute the Legendre symbol only if the modulus is a prime number. The following table demonstrates the dependency between the value of the Legendre symbol and solvability of the congruence:
If the Legendre number is... | The congruence... |
---|---|
1 | Has one or more solutions |
0 | Cannot be solved by numlib::msqrts. Try numlib::mroots. |
-1 | Has no solution |
MuPAD^{®} implements the Legendre symbol as the numlib::legendre function. If, and only if, the congruence x^{2}≡ a(mod m) is solvable, the Legendre symbol is equal to 1:
numlib::legendre(12, 13)
numlib::msqrts(12, 13)
If, and only if, the congruence x^{2}≡ a(mod m) does not have any solutions, the Legendre symbol is equal to -1:
numlib::legendre(11, 13)
numlib::msqrts(11, 13)
If a and m are not coprime, the Legendre symbol is equal to 0. In this case, numlib::legendre function returns 0, and numlib::msqrts errors:
numlib::legendre(13, 13)
numlib::msqrts(13, 13)
Error: Arguments must be relative prime. [numlib::msqrts]
You can compute the Legendre symbol only if the modulus is a prime number. If a congruence has a nonprime odd modulus, you can compute the Jacobi symbol. The Jacobi symbol determines the unsolvable congruences x^{2}≡ a(mod m). You cannot compute the Jacobi symbol if the modulus is an even number. The following table demonstrates the dependency between the value of the Jacobi symbol and the solvability of the congruence:
If the Jacobi number is... | The congruence... |
---|---|
1 | Might have solutions |
0 | Cannot be solved by numlib::msqrts. Try numlib::mroots. |
-1 | Has no solutions |
MuPAD implements the Jacobi symbol as the numlib::jacobi function. If the Jacobi symbol is equal to -1, the congruence does not have a solution:
numlib::jacobi(19, 21)
numlib::msqrts(19, 21)
If Jacobi symbol is equal to 1, the congruence might have solutions:
numlib::jacobi(16, 21)
numlib::msqrts(16, 21)
However, the value 1 of the Jacobi symbol does not guarantee that the congruence has solutions. For example, the following congruence does not have any solutions:
numlib::jacobi(20, 21)
numlib::msqrts(20, 21)
If a and m are not coprime, the Jacobi symbol is equal to 0. In this case, numlib::jacobi function returns 0, and numlib::msqrts errors:
numlib::jacobi(18, 21)
numlib::msqrts(18, 21)
Error: Arguments must be relative prime. [numlib::msqrts]
Besides solving a linear congruence or computing modular square roots, MuPAD also enables you to solve congruences of a more general type of P(x) ≡ 0(mod m). Here P(x) is a univariate or multivariate polynomial. To solve such congruences, use the numlib::mroots function. For example, solve the congruence x^{3}+ x^{2}+ x + 1 ≡ 0(mod 3). First, define the left side of the congruence as a polynomial by using the poly function:
p := poly(x^3 + x^2 + x + 1)
Now, use the numlib::mroots function to solve the congruence:
numlib::mroots(p, 299)
Using the numlib::mroots function, you also can solve the congruence for a multivariate polynomial. For a multivariate polynomial P( x_{1}, ..., x_{n}), numlib::mroots returns a nested list as a result. Each inner list contains one solution x_{1}, ..., x_{n}. For example, find modular roots of the following multivariate polynomial:
p := poly(x^3*y^2 + x^2*y + x + y + 1): numlib::mroots(p, 11)