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Summary
This summary is machine-generated.

This study analyzes the Diophantine equation U_n = p^x for linear recurrence sequences. For most prime numbers p, there is at most one solution (n, x), with exceptions computed for specific sequences.

Keywords:
Diophantine equationsExponential Diophantine equationsLinear recurrence sequences

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Area of Science:

  • Number Theory
  • Diophantine Equations
  • Recurrence Relations

Background:

  • Diophantine equations involving linear recurrence sequences are a rich area of number theory.
  • Understanding the solutions to U_n = p^x provides insights into the distribution of prime powers within these sequences.

Purpose of the Study:

  • To investigate the number of solutions to the Diophantine equation U_n = p^x, where U_n is a linear recurrence sequence and p is a prime.
  • To determine conditions under which at most one solution exists for a given prime p.
  • To compute the set of exceptional primes for which more than one solution might exist for specific sequences.

Main Methods:

  • The study employs techniques from the theory of Diophantine equations and linear recurrence sequences.
  • It establishes bounds and conditions based on the properties of the sequence U_n.
  • Computational methods are used to identify specific exceptional sets for the Tribonacci and Lucas sequences.

Main Results:

  • For linear recurrence sequences satisfying certain hypotheses, the equation U_n = p^x has at most one solution (n, x) for almost all prime numbers p.
  • An effectively computable finite set of exceptional primes is identified.
  • The exceptional sets for the Tribonacci sequence and the Lucas sequence plus one are explicitly computed.

Conclusions:

  • The research significantly narrows down the possible solutions for this type of Diophantine equation.
  • The findings contribute to the understanding of prime powers within linear recurrence sequences.
  • The explicit computation of exceptional sets for specific sequences provides valuable data for further theoretical and computational number theory research.