Department of Mathematics, Faculty of Science, Okayama University
Doi
Abstract
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
