A new family of elliptic curves with positive ranks arising from the Heron triangles

The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$. It turned out that the torsion groups of this family are of the form $\frac{\Bbb{Z}}{2\Bbb{Z}}\times \frac{\Bbb{Z}}{2\Bbb{Z}}$ and also the rank is positive

Sums of two biquadrates and elliptic curves of rank ≥4\geq 4

If an integer $n$ is written as a sum of two biquadrates in two different ways, then the elliptic curve $y^2=x^3-nx$ has rank $\geq 3$. If moreover $n$ is odd and the parity conjecture is true, then it has even rank $\geq 4$. Finally, some examples of ranks equal to 4, 5, 6, 7, 8 and 10, are also obtained.Comment: 11 pages, 2 table

SUMS OF TWO BIQUADRATES AND ELLIPTIC CURVES OF RANK ≥ 4

If an integer n is written as a sum of two biquadrates in two different ways, then the elliptic curve y2 = x3 − nx has positive rank. We utilize Euler’s parametrization to introduce some homoge- neous equations to prove that En has rank ≥ 3. If moreover n is odd and the parity conjecture is true, then the curve has even rank ≥ 4. Finally, some examples of ranks equal to 4, 5, 6, 7, 8 and 10, are also obtained