The 2-Adic Valuation of Shifted Padovan and Perrin Numbers and Applications

Abstract

We characterize the 2-adic valuation of (Pn - 1) n >= 0 , where ( P n ) n >= 0 denotes the Padovan sequence. In addition, we use this formula to find all the Cullen and Proth numbers that are Padovan numbers. We also fully describe the 2-adic order of (Rn + 1) n >= 0 , where ( R n ) n >= 0 denotes the Perrin sequence, and use it to find all Woodall and Proth numbers of the second kind which are Perrin numbers. As a consequence we find that 3 , 5 , 9 , and 65 are the only Fermat numbers in the Padovan sequence; while 3 and 7 and 2 and 5 are the only numbers of Mersenne and Th & acirc;bit ibn Kurrah in the Perrin sequence respectively.