TY - JOUR
T1 - Partitions with equal products and elliptic curves
AU - Sadek, Mohammad
AU - El-Sissi, Nermine
N1 - Publisher Copyright:
© 2015, Osaka University. All rights reserved.
PY - 2015
Y1 - 2015
N2 - Let a, b, c be distinct positive integers. Set M = a + b + c and N = abc. We give an explicit description of the Mordell-Weil group of the elliptic curve E(M,N): y2-Mxy-Ny = x3 over ℚ. In particular we determine the torsion subgroup of E(M,N)(ℚ) and show that its rank is positive. Furthermore there are infinitely many positive integers M that can be written in n different ways, n ∈ {2, 3}, as the sum of three distinct positive integers with the same product N and E(M,N)(ℚ) has rank at least n.
AB - Let a, b, c be distinct positive integers. Set M = a + b + c and N = abc. We give an explicit description of the Mordell-Weil group of the elliptic curve E(M,N): y2-Mxy-Ny = x3 over ℚ. In particular we determine the torsion subgroup of E(M,N)(ℚ) and show that its rank is positive. Furthermore there are infinitely many positive integers M that can be written in n different ways, n ∈ {2, 3}, as the sum of three distinct positive integers with the same product N and E(M,N)(ℚ) has rank at least n.
UR - http://www.scopus.com/inward/record.url?scp=84925424182&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84925424182
SN - 0030-6126
VL - 52
SP - 515
EP - 525
JO - Osaka Journal of Mathematics
JF - Osaka Journal of Mathematics
IS - 2
ER -