NishMath

Recent research in number theory is focusing on the presence of perfect powers within the Lucas and Fibonacci sequences. A perfect power is a number that can be expressed as an integer raised to an integer power, like 4 (2^2) or 8 (2^3). The study aims to identify and prove conjectures related to perfect powers within these sequences, with initial findings suggesting such numbers are sparse. For the Fibonacci sequence, previous work has shown the only perfect powers to be 0, 1, 8, and 144 (0, 1, 2^3, and 12^2 respectively). For the Lucas sequence, only 1 and 4 (1 and 2^2 respectively) are perfect powers.

A related line of inquiry involves examining products of terms from these sequences. A conjecture suggests that 2^4 is the only perfect power of the form F_m * F_n, and it is also conjectured that L_0 * L_3, L_0 * L_6 and L_1 * L_3 are the only perfect powers of the form L_m * L_n with specific limits placed on their indices. Additionally, researchers are investigating a diophantine equation of the form (2^m ± 1)(2^n ± 1) = x^k, and attempting to establish that (2^3-1)(2^6-1)=21^2 is the only perfect power of the form (2^m -1)(2^n - 1), while (2+1)(2^3+1)=3^3 is the only perfect power of the form (2^m + 1)(2^n + 1).

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References:

• Recent Questions - Mathematics Stack Exchange - Which Lucas numbers are perfect powers? - 6d
• doi.org - Perfect powers as products of two Fibonacci or Lucas numbers - 5d
• Recent Questions - MathOverflow - Perfect powers as products of two Fibonacci or Lucas numbers - 5d

Classification:

• HashTags: NumberTheory DiophantineEquations PerfectPowers
• Company: Math
• Target: Equations
• Product: NumberTheory
• Feature: NumberTheory
• Type: Research
• Severity: Major