A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets (a,b,c)∈N3*, we give a new proof of the well-known problem of these particular squareless numbers n∈N∗
, called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: (Aα)2(Bβ)2=(Cγ)2
, such that its area Δ=12AαBβ=n
; or in an equivalent way, to that of the existence of numbers U2 ,V2 ,W2 ∈Q2* that are in an arithmetic progression of reason n; Problem equivalent to the existence of: (a,b,c)∈N3* prime in pairs, and f ∈N* , such that: (a−b2f)2
, (c2f)2
, (ab2f)2
are in an arithmetic progression of reason n ; And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a2 + b2 = c2, such that its area Δ=12ab=nf2
, where f ∈N*, and this last equation can be written as follows...