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TWO PARAMETRIC FAMILIES OF CONGRUENT NUMBERS FROM ELLIPTIC CURVES WITH QUADRATIC POINT IMPOSITION
Keywords:
congruent numbers, elliptic curves, positive rank, quadratic points, parametric familiesDOI:
https://doi.org/10.17654/0972555525043Abstract
We introduce two infinite parametric families of congruent numbers constructed from elliptic curves by imposing a rational point with coordinate $x=t^2$. The first family,
$$
m_1(a, b)=a^4+3 a^2 b^2+2 b^4
$$
arises from the elliptic curve
$$
E(t): y^2=x^3-(2 t+1)^2 x,
$$
while the second,
$$
n_2(a, b)=\left|a^4-3 a^2 b^2+2 b^4\right|
$$
comes from the curve
$$
E(t): y^2=x^3-\left(t^2-1\right)^2 x
$$
We present algebraic derivations, provide a heuristic confirmation of congruence via computations of the associated elliptic curves, and deliver a comparative statistical analysis of these families over a large integer range. Our results illustrate the effectiveness of quadratic point impositions in generating rich and diverse congruent number sets, advancing both theoretical understanding and computational explorations of the classical congruent number problem.
Received: September 7, 2025
Accepted: October 29, 2025
References
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