Product of Two Consecutive Fibonacci or Lucas Numbers Divisible by their Prime Sum of Indices
Vladimir Pletser
TL;DR
This paper proves that the product of two consecutive Fibonacci or Lucas numbers is divisible by the sum of their indices when that sum is a prime of specific forms, expanding understanding of divisibility properties in these sequences.
Contribution
It establishes new divisibility criteria for Fibonacci and Lucas numbers based on prime sums of their indices, with conditions on the form of the primes.
Findings
Divisibility occurs when the sum of indices is a prime of form (4r+1) for Fibonacci.
Divisibility occurs when the sum of indices is a prime of form (4r+3) for Lucas.
The prime 5 is excluded from the divisibility criteria.
Abstract
We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Identities · Advanced Mathematical Theories