Scientific foundation

A continuous research path from fixed-cathetus triples to elliptic curves.

The technology is grounded in Roberto Amato's published work on divisor-based Pythagorean parametrization, primitive triples, algebraic structures, prime characterization and congruent-number elliptic curves.

Divisor parameter and inradius

The elementary variables behind the construction.

Let x > 1 be an odd integer. For this odd case, an admissible parameter d is a positive divisor of x² satisfying d < x. Each such divisor determines a nondegenerate integral right triangle with fixed odd leg x:

y = \frac{x^{2}}{2 d} - \frac{d}{2}

z = \frac{x^{2}}{2 d} + \frac{d}{2}

Here d is not an arbitrary geometric length: it is the divisor parameter selecting a member of the fixed-cathetus Pythagorean family. The inradius r of that triangle satisfies

r = \frac{x - d}{2}, hence x = 2 r + d

In the canonical specialization d = 1, this becomes

x = 2 r + 1

This is the relation used by the Pythagorean Sieve.

Congruent-number curve

From the triangle to an explicit elliptic-curve point.

The area N of a rational right triangle leads, through the classical triangle–curve correspondence, to a rational point on the congruent-number elliptic curve

E_{N} : Y^{2} = X^{3} - N^{2} X

The divisor-based parametrization therefore associates admissible pairs (x,d) with explicit pointed elliptic curves. For odd x, the uniqueness of the admissible divisor d = 1 is equivalent to the corresponding elementary primality criterion.

Research thread

Geometry, divisors, primes and rational points.

Fixed cathetus

A divisor of the square of a prescribed leg determines the corresponding integral Pythagorean triangle.

Inradius index

The relation between the divisor parameter and the inradius yields the arithmetic indexing used by the sieve.

Prime uniqueness

For odd x, primality becomes a uniqueness statement for the admissible divisor family.

Elliptic correspondence

Explicit Pythagorean triangles lead to distinguished rational and integral points on congruent-number curves.

Selected number-theory publications

Published research and scientific records.

2026Pythagorean Divisor Families and Explicit Rational Points on Congruent-Number Elliptic CurvesZenodo publication record · DOI 10.5281/zenodo.22081722 2025A Pythagorean Sieve and a Recursive Modular Characterization of Odd PrimesInternational Journal of Applied Mathematics 2025Characterization of Diophantine Equations a + y² = z², Pythagorean n-Tuples, and Algebraic StructuresInternational Journal of Mathematics and Mathematical Sciences 2024A Novel Approach for Studying Pythagorean Triples Suitable for Students at All Educational LevelsEuropean Journal of Pure and Applied Mathematics 2024Groups and Monoid in the Set of Pythagorean TriplesIntegers 2023A Characterization of Primitive Pythagorean TriplesPalestine Journal of Mathematics 2021Some Relations Among Pythagorean TriplesInternational Journal of Mathematics and Computer Science 2020A Note on Pythagorean TriplesInternational Journal of Mathematics and Computer Science 2017A Characterization of Pythagorean TriplesJP Journal of Algebra, Number Theory and Applications

Public records

Reproducibility without disclosure of the proprietary core.

Pythagorean Sieve public computational record10.5281/zenodo.21983583 ECC computational verification record10.5281/zenodo.21907480

Technical brochure

Review the RSA-oriented and ECC-oriented technologies in one concise document.

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