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Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b
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Let (Pk)k≥0 be a Padovan sequence and (Rk)k≥0 be a Perrin sequence. Let n, m, b, and k be non-negative integers, where 2≤b≤10. In this paper, we are devoted to delving into the equations Rn=bdPm+Rk and Rn=bdRm+Pk, where d is the number of digits of Rk or Pk in base b. We show that the sets of solutions are Rn∈{R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R19,R23,R25,R27} for the first equation and Rn∈{R0,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R18,R20,R21} for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b. We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences.
Title: Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b
Description:
Let (Pk)k≥0 be a Padovan sequence and (Rk)k≥0 be a Perrin sequence.
Let n, m, b, and k be non-negative integers, where 2≤b≤10.
In this paper, we are devoted to delving into the equations Rn=bdPm+Rk and Rn=bdRm+Pk, where d is the number of digits of Rk or Pk in base b.
We show that the sets of solutions are Rn∈{R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R19,R23,R25,R27} for the first equation and Rn∈{R0,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R18,R20,R21} for the second equation.
Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b.
We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences.
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