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In this paper we show that for any two primes p and q greater than 5, theelliptic curve E(p,q) : y2 = x3 − p2x + q2 has rank at least 2. We will also provide twoindependent points on E(p,q). Then we will show that, conjecturally, the family {E(p,q)}contains an infinite subfamily of rank three elliptic curves.

facta universitatis. series: mathematics and informatics

FAMILY OF ELLIPTIC CURVES &lt;em&gt;E&lt;sup&gt;&lt;/em&gt;(&lt;em&gt;p&lt;/em&gt;,&lt;em&gt;q&lt;/em&gt;)&lt;/sup&gt;‎: &lt;em&gt;‎y&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt;=&lt;em&gt;x&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt;-&lt;em&gt;p&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;em&gt;x&lt;/em&gt;+&lt;em&gt;q&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt;