Impossible rational triangles

A rational triangle is a triangle whose sides have rational length and whose area is rational. Can any two rational numbers be sizes of a rational triangle? Surprisingly no. You can always find a third side of rational length, but it might not be possible to do so while keeping the area rational.

The following theorem comes from [1].

Let q be a prime number not congruent to 1 or −1 mod 8. And let ω be an odd integer such that no prime factor of q^2ω − 1 is congruent to 1 mod 4. Then a = q^ω + 1 and b = q^ω − 1 are not the sides of any rational triangle.

To make the theorem explicitly clear, here’s a Python implementation.

from sympy import isprime, primefactors

def test(q, omega):
    if not isprime(q):
        return False
    if q % 8 == 1 or q % 8 == 7:
        return False
    if omega % 2 == 0:
        return False
    factors = primefactors(q**(2*omega) - 1)
    for f in factors:
        if f % 4 == 1:
            return False
    return True

We can see that q = 2 and ω = 1 passes the test, so there is no rational triangle with sides a = 2¹ + 1 = 3 and b = 2¹ − 1 = 1.

You could use the code above to search for (a, b) pairs systematically.

from sympy import primerange

for q in primerange(20):
    for omega in range(1, 20, 2):
        if test(q, omega):
            a, b = q**omega + 1, q**omega - 1
            print(a, b)

This outputs the following:

3 1
9 7
33 31
129 127
8193 8191
32769 32767
131073 131071
524289 524287
4 2
6 4
126 124
14 12

Note that one of the outputs is (4, 2). Since multiplying the sides of a rational triangle by a rational number gives another rational triangle, there cannot be a rational triangle in which one side is twice as long as another.

As the author in [1] points out, “There are undoubtedly many pairs not covered by [the theorem cited above]. It would be of interest to characterize all pairs.” The paper was published in 1976. Maybe there has been additional work since then.

[1] N. J. Fine. On Rational Triangles. The American Mathematical Monthly. Vol. 83. No. 7, pp. 517–521.

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