The Probability of Relatively Prime Polynomials

  title={The Probability of Relatively Prime Polynomials},
  author={Arthur T. Benjamin and Curtis D. Bennett},
  journal={Mathematics Magazine},
  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 
The probability of relatively prime polynomials in ℤpk[x]
Let PR(m, n) denote the probability that two randomly chosen monic polynomials f , g ∈ R[x] of degrees m and n, respectively, are relatively prime. Let q = p, be a prime power. We establish an
On the Number of Factorizations of Polynomials over Finite Fields
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  • Mathematics, Computer Science
    2020 IEEE International Symposium on Information Theory (ISIT)
  • 2020
For the two enumeration problems, bounds are obtained on the maximum number of factorizations, and a characterization is presented for polynomials attaining that maximum.
Recursive sequences and polynomial congruences
We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer m. We show that the period of
A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields
Let f ( x ) = x n + ( a n − 1 t + b n − 1 ) x n − 1 + · · · + ( a 0 t + b 0 ) be of constant degree n in x and degree ≤ 1 in t , where all a i , b i are randomly and uniformly selected from a finite
On multiplicative independence of rational function iterates
  • M. Young
  • Mathematics
    Monatshefte für Mathematik
  • 2020
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of
On Certain Probabilistic Properties of Polynomials over the Ring of p-adic Integers
The probability that a monic polynomial is separable, generalizing a result of Polak is calculated and the notion of two polynomials being strongly coprime is introduced and the method can be used to extrapolate other probabilistic properties of polynmials over the ring of p-adic integers.


A Pentagonal Number Sieve
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
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