Congruum

Find an integer a, and x,y,z rational with :
y² - a = x²
y² + a = z²
Number a is named a "congruent" number, or congruum.

Congruent squares

The problem is finding x, y, z integers
Adding, we get 2y² = x² + z²
x and z have then same parity so we can define z + x = 2u, z - x = 2v. That is x = u - v and z = u + v.
2y² = (u + v)² + (u - v)² = 2(u² + v²) hence y² = u² + v².
u,v,y are then a Pythagorean triple, given by generators :
u = 2mpq, v = m(p² - q²), y = m(p² + q²) (or exchanging u and v).
As 2a = z² - x² = (z + x)(z - x) = 4uv : a = 4m²pq(p² - q²)

We get then all solutions by m,p,q ∈ Z in the relations :

 a = 4m²pq(p² - q²) x = m|2pq - (p² - q²)| y = m(p² + q²) z = m(2pq + (p² - q²))

Congruum

That is find an integer a = 4m²pq(p² - q²)/k²
If a is a congruent number, also k²a and we care only of those "square free" a = pq(p² - q²)/k²
A congruent number is also the area of some right triangle with rational sides 2pq/k², (p² - q²)/k², (p² + q²)/k²

It is not obvious if a given number, even a small one, is of that form or not, may be with huge values of k,p and q.

axyz
540/69/641/6
6435
7288/60175/60337/60
(more values require Javascript).

Extending the search with much higher values for p and q, we find that the only square free congruent numbers <100 are :
5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95.
For instance 13 is a congruent number, area of a right triangle with sides (at simplest) :
23400/9690 = 780/323, 104329/9690 = 323/30, 106921/9690, obtained for p = 325 q = 36 .

Knowing in advance if a number is congruent or not is a difficult problem. Just a few conditions

• A prime of the form 8k + 3 is never a congruent number
• A 2p number with p = 8k + 5 prime is never a congruent number
• A number pq with p and q primes of the form 8k + 3 is not congruent
• A number 2pq with p and q primes of the form 8k + 5 is not congruent

At last, the modern way is to consider the problem as searching rational points on the elliptic curve Y² = X(X² - a²)

A table of congruent numbers with corresponding values could be found here (Japanese site).
Note some "monsters" : a = 53, p = 1873180325, q = 1158313156 where p and q are huge compared to a, but the next one a = 55, p = 125, q = 44 only !