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Added code to calculate crt for non-coprime moduli in Integer #39716

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Added code to calculate crt for non-coprime moduli in Integer #39716

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7ad3e06
Added code to calculate crt for non-coprime moduli in Integer
Mar 16, 2025
c959367
Fixed documentation that said n and m must be coprime.
Mar 16, 2025
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16 changes: 13 additions & 3 deletions src/sage/rings/integer.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -6965,8 +6965,6 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
Return the unique integer between `0` and `mn` that is congruent to
the integer modulo `m` and to `y` modulo `n`.

We assume that `m` and `n` are coprime.

EXAMPLES::

sage: n = 17
Expand All @@ -6976,6 +6974,16 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
17
sage: m%11
5

`crt` also works for some non-coprime moduli::

sage: 6.crt(0,10,4)
16
sage: 6.crt(0,10,10)
Traceback (most recent call last):
...
ValueError: no solution to crt problem since gcd(10,10) does not
divide 6-0
"""
cdef object g, s
cdef Integer _y, _m, _n
Expand All @@ -6984,7 +6992,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
_n = Integer(n)
g, s, _ = _m.xgcd(_n)
if not g.is_one():
raise ArithmeticError("CRT requires that gcd of moduli is 1.")
if (self % g) <> (_y % g):
raise ValueError("no solution to crt problem since gcd(%s,%s) does not divide %s-%s" % (_m, _n, self, _y))
return (self + g * Integer(0).crt((_y - self) // g, _m // g, _n // g)) % _n.lcm(_m)
# Now s*m + t*n = 1, so the answer is x + (y-x)*s*m, where x=self.
return (self + (_y - self) * s * _m) % (_m * _n)

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