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132 lines
3.1 KiB
Python
132 lines
3.1 KiB
Python
from typing import Tuple
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def diophantine(a: int, b: int, c: int) -> Tuple[float, float]:
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"""
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Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
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diophantine equation a*x + b*y = c has a solution (where x and y are integers)
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iff gcd(a,b) divides c.
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GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
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>>> diophantine(10,6,14)
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(-7.0, 14.0)
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>>> diophantine(391,299,-69)
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(9.0, -12.0)
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But above equation has one more solution i.e., x = -4, y = 5.
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That's why we need diophantine all solution function.
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"""
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assert (
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c % greatest_common_divisor(a, b) == 0
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) # greatest_common_divisor(a,b) function implemented below
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(d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
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r = c / d
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return (r * x, r * y)
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def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
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"""
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Lemma : if n|ab and gcd(a,n) = 1, then n|b.
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Finding All solutions of Diophantine Equations:
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Theorem : Let gcd(a,b) = d, a = d*p, b = d*q. If (x0,y0) is a solution of
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Diophantine Equation a*x + b*y = c. a*x0 + b*y0 = c, then all the
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solutions have the form a(x0 + t*q) + b(y0 - t*p) = c,
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where t is an arbitrary integer.
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n is the number of solution you want, n = 2 by default
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>>> diophantine_all_soln(10, 6, 14)
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-7.0 14.0
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-4.0 9.0
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>>> diophantine_all_soln(10, 6, 14, 4)
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-7.0 14.0
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-4.0 9.0
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-1.0 4.0
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2.0 -1.0
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>>> diophantine_all_soln(391, 299, -69, n = 4)
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9.0 -12.0
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22.0 -29.0
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35.0 -46.0
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48.0 -63.0
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"""
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(x0, y0) = diophantine(a, b, c) # Initial value
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d = greatest_common_divisor(a, b)
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p = a // d
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q = b // d
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for i in range(n):
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x = x0 + i * q
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y = y0 - i * p
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print(x, y)
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def greatest_common_divisor(a: int, b: int) -> int:
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"""
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Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
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Euclid's Algorithm
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>>> greatest_common_divisor(7,5)
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1
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Note : In number theory, two integers a and b are said to be relatively prime,
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mutually prime, or co-prime if the only positive integer (factor) that
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divides both of them is 1 i.e., gcd(a,b) = 1.
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>>> greatest_common_divisor(121, 11)
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11
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"""
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if a < b:
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a, b = b, a
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while a % b != 0:
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a, b = b, a % b
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return b
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def extended_gcd(a: int, b: int) -> Tuple[int, int, int]:
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"""
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Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers
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x and y, then d = gcd(a,b)
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>>> extended_gcd(10, 6)
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(2, -1, 2)
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>>> extended_gcd(7, 5)
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(1, -2, 3)
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"""
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assert a >= 0 and b >= 0
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if b == 0:
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d, x, y = a, 1, 0
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else:
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(d, p, q) = extended_gcd(b, a % b)
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x = q
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y = p - q * (a // b)
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assert a % d == 0 and b % d == 0
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assert d == a * x + b * y
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return (d, x, y)
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if __name__ == "__main__":
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from doctest import testmod
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testmod(name="diophantine", verbose=True)
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testmod(name="diophantine_all_soln", verbose=True)
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testmod(name="extended_gcd", verbose=True)
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testmod(name="greatest_common_divisor", verbose=True)
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