Congruent Numbers

Over the years I've sometimes made note of when a number associated with my diurnal age is congruent. Often I've just ignored the fact. The definition of such number is:

A number is called congruent it is the area of a right triangle with rational sides.

The first congruent number is 5 because it is the area of a right triangle with sides of 20/3, 3/2, and 41/6. The next number, 6, is also congruent being the area of the famous 3, 4, 5 right triangle. See Figure 1.

Figure 1

7 is also a congruent number being the area of a right triangle with sides such that (see Figure 2):$$ \begin{align} \Big ( \frac{35}{12} \Big )^2+ \Big ( \frac{24}{5} \Big )^2 &= \Big ( \frac{337}{60} \Big )^2 \\ \frac{1}{2} \cdot \frac{35}{12} \cdot \frac{24}{5} &= 7 \end{align}$$

Figure 2

With 7, we begin to see the problem of determining whether a number is congruent or not. How do we find those fractions that confirm that 7 is congruent. It's not easy. To quote from Wikipedia:

The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2019) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Without getting too deeply into this topic, it suffices to say that OEIS A006991 provides a list of so-called primitive congruent numbers up to 10,000.  These are congruent numbers that are square-free. If any of these numbers are multiplied by a positive square number, a new congruent number is created.

Let's take the number associated with my diurnal age yesterday as an example. 27260 factorises as follows:$$ \begin{align} 27260 &= 2^2 \times 5 \times 29 \times 47\\ &= 2^2 \times 6815 \end{align}$$Now 6815 is a member of OEIS A006991 and therefore 27260 is thus a congruent number because a primitive congruent number, 6815, has been multiplied by a positive square number (4).

It can be noted that 6814 is also a member of OEIS A006991 and thus 4 x 6814 = 27256 is also a congruent number. It's not uncommon for primitive congruent numbers to occur in pairs or even triples.  For example: (5, 6, 7), (13, 14, 15), (21, 22, 23), (29, 30, 31), 34 is a singleton, (37, 38, 39) etc.