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CoprimeQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

CoprimeQ[n1, n2, ...]
    yields True if the arguments are pairwise relatively prime, and
    False otherwise.
Integers are relatively prime when their GCD is 1.  Works for machine
and BigInt integers.  With GaussianIntegers -> True, or when any
argument is an exact Gaussian integer, coprimality is tested over the
Gaussian integers Z[i].  Returns False unless the arguments are
manifestly coprime; CoprimeQ[] is False and CoprimeQ[n] is True.
Listable and Orderless.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= CoprimeQ[8, 11]
Out[1]= True

In[2]:= CoprimeQ[2, 4]
Out[2]= False

In[3]:= CoprimeQ[2, 3, -5, 7]
Out[3]= True

In[4]:= CoprimeQ[5 + I, 1 - I]
Out[4]= False

In[5]:= CoprimeQ[{1, 2, 3, 4, 5}, 6]
Out[5]= {True, False, False, False, True}

Implementation notes

  • Machine integers and GMP bigints, handled uniformly through mpz_gcd, so large cases are exact: CoprimeQ[2^100 - 1, 3^100 - 1] → False (both even), CoprimeQ[2^127 - 1, 2^61 - 1] → True. Sign is ignored; GCD(0, n) = |n|, so CoprimeQ[0, 1] → True but CoprimeQ[0, 5] → False.
  • More than two arguments are tested pairwise: CoprimeQ[6, 35, 143] → True, while CoprimeQ[2, 3, 4] → False (2 and 4 share a factor).
  • Gaussian integers: with GaussianIntegers -> True, or when any argument is an exact Gaussian integer, coprimality is tested over Z[i] via the Gaussian Euclidean algorithm (round-to-nearest division). CoprimeQ[5 + I, 1 - I] → False (both divisible by 1 + I); CoprimeQ[2, 5, GaussianIntegers -> True] → True, while CoprimeQ[2, 10, GaussianIntegers -> True] → False.
  • Orderless: argument order is irrelevant, and the GaussianIntegers option may appear at any position.
  • Listable: threads element-wise over lists, e.g. CoprimeQ[{1, 2, 3, 4, 5}, 6] → {True, False, False, False, True}.
  • As a *Q predicate it always returns a Boolean: CoprimeQ[] → False, CoprimeQ[n] → True (no pairs), and anything not a manifestly coprime integer or Gaussian integer — rationals, reals, symbols, malformed options — yields False (e.g. CoprimeQ[a, b] → False).

Attributes: Listable, Orderless, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References