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.