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Let p be a fixed prime and k be a fixed odd positive integer. Further let N(pk) denote the number of pairs of integer points (x,±y) on the elliptic curve E:y2=x3−pkx with y>0. Using some properties of the Diophantine equations, we gave an exact upper bound estimate for N(pk). That is, we proved that N(pk)≤4.

An exact upper bound estimate for the number of integer points on the elliptic curves y 2 = x 3 − p k x