The Probability of Relatively Prime Polynomials

@article{Benjamin2007ThePO,
  title={The Probability of Relatively Prime Polynomials},
  author={Arthur T. Benjamin and Curtis D. Bennett},
  journal={Mathematics Magazine},
  year={2007},
  volume={80},
  pages={196 - 202}
}
The one sentence proof is that any number that divides a and b must also divide b and r (since r = a ? qb) and vice versa; hence, the pairs (a, b) and (b, r) have the exact same set of common divisors. What turns this theorem into an algorithm is that if b > 0, then we can find a unique quotient q so that 0 < r < b, allowing us to repeat the process with the second coordinate decreasing to zero. That is, if gcd(a, b) = c, then Euclid's algorithm will look like 
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References

SHOWING 1-3 OF 3 REFERENCES
A Pentagonal Number Sieve
TLDR
A general “pentagonal sieve” theorem is proved that has corollaries such as the following: iff,gare two monic polynomials of the same degree over the fieldGF(q), then the probability thatf,Gare relatively prime is 1?1/q.
On an Involution Concerning Pairs of Polynomials over F2
TLDR
The number of coprime m-tuples of monic polynomials of degree n over Fq is equal to qnm?q(n?1)m+1 and an involution is given that proves this result.
On an Involution Concerning Pairs of Polynomials in F 2
  • Journal of Combinatorial Theory, Series A
  • 2000