Let \(n\) be a positive integer and \(E_n\) be an elliptic curve defined by \(y^2=x(x^2-n^2)\). This curve is called a congruent number elliptic curve since \(n\) is a congruent number if and only if it has a rational points other than \(2\)-division points. If \(P_i,i=1,2,3,\dots,m\), are rational points on an elliptic curve, then they form an arithmetic progression (of length \(m\)) if their \(x\)-coordinates form an arithmetic progression. A. Bremner [Exp. Math. 8, No. 4, 409–413 (1999; Zbl 0951.11021)] noted that rational points in nontrivial arithmetic progression tend to be independent. This implies the rank of the curve with a non-trivial arithmetic progression of length \(m\) is likely to be at least \(m\). The author constructs infinitely many curves \(E_n\) having a nontrivial arithmetic progression of length \(3\) consisted of integral independent points, from the rational points of the curve \(w^2=9t^4+4t^2+36\).