A criterion for stability of two-term recurrence sequences modulo \(2^k\) (Q1383510)

Let \(a\) and \(b\) be fixed integers and let \(\{u_i\), \(i\geq 0\}\) be the two-term recurrence sequence defined by \(u_0=0\), \(u_1=1\), and for all \(i\geq 2: u_i= au_{i-1}+ bu_{i-2}\). A new and interesting technique for characterizing stable sequences of the above type is described, where \(a\) is odd, \(b\equiv 3\pmod 4\). This technique is applied to identify stable sequences that were not previously known to be stable.