Abstract
AbstractThe integer sequence defined by$$P_{n+3}=P_{n+1}+P_{n}$$Pn+3=Pn+1+Pnwith initial conditions$$P_{0}=P_{1}=P_{2}=1$$P0=P1=P2=1is known as the Padovan sequence$$(P_{n})_{n\in \mathbb {Z}}$$(Pn)n∈Z. The Perrin sequence$$(R_{m})_{m\in \mathbb {Z}}$$(Rm)m∈Zsatisfies the same recurrence equation as the Padovan sequence but with starting values$$R_{0}=3$$R0=3,$$R_{1}=0$$R1=0, and$$R_{2}=2$$R2=2. In this note, we solve the Diophantine equation$$P_{n}=\pm R_{m}$$Pn=±Rmwith$$(n,m)\in \mathbb {Z}^{2}$$(n,m)∈Z2.
Publisher
Springer Science and Business Media LLC
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