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.
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))
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.
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 curveY² = 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 !