Suppose you fill two n×n matrices with random integers. What is the probability that the determinants of the two matrices are relatively prime? By “random integers” we mean that the integers are chosen from a finite interval, and we take the limit as the size of the interval grows to encompass all integers.
Let Δ(n) be the probability that two random integer matrices of size n have relatively prime determinants. The function Δ(n) is a strictly decreasing function of n.
The value of Δ(1) is known exactly. It is the probability that two random integers are relatively prime, which is well known to be 6/π². I’ve probably blogged about this before.
The limit of Δ(n) as n goes to infinity is known as the Hafner-Sarnak-McCurley constant , which has been computed to be 0.3532363719…
Since Δ(n) is a decreasing function, the limit is also a lower bound for all n.
Here is some Python code to experiment with the math discussed above. We’ll first do a simulation to show that we get close to 6/π² for the proportion of relatively prime pairs of integers. Then we look at random 2×2 determinants.
from sympy import gcd from numpy.random import randint from numpy import pi def coprime(a, b): return gcd(a, b) == 1 def random_int(N): return randint(-N, N) def random_det(N): a, b, c, d = randint(-N, N, 4) return a*d - b*c count = 0 N = 10000000 # draw integers from [-N, N) num_reps = 1000000 for _ in range(num_reps): count += coprime(random_int(N), random_int(N)) print("Simulation: ", count/num_reps) print("Theory: ", 6*pi**-2)
This code printed
Simulation: 0.607757 Theory: 0.6079271018540267
when I ran it, so our simulation agreed with theory to three figures, the most you could expect from 106 repetitions.
The analogous code for 2×2 matrices introduces a function
def random_det(N): a, b, c, d = randint(-N, N, 4, dtype=int64) return a*d - b*c
I specified the
dtype because the default is to use (32 bit)
int as the type, which lead to Python complaining “RuntimeWarning: overflow encountered in long_scalars”.
random_det and reran the code above. This produced 0.452042. The exact value isn’t known in closed form, but we can see that it is between the bounds Δ(1) = 0.6079 and Δ(∞) = 0.3532.
In  the authors show that
This expression is only known to have a closed form when n = 1.
 Hafner, J. L.; Sarnak, P. & McCurley, K. (1993), “Relatively Prime Values of Polynomials”, in Knopp, M. & Seingorn, M. (eds.), A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., ISBN 0-8218-5155-1.