When Fibonacci and Lucas Meet Smith: A Computational Exploration
Abstract
The Fibonacci and Lucas sequences are old friends in recreational number theory. Here we ask a simple question: which of their terms Fn and Ln are Smith numbers? A Smith number is a composite integer whose decimal digit sum equals the sum of the decimal digits of its prime factors, counted with multiplicity. The question is easy to state, but for large n it quickly becomes a factorization problem. Using available complete factorization data, we carried out computational searches in both sequences and obtained new Smith terms in each of them. For the Lucas sequence, the first author's search found ten Smith terms with indices below 1000: 3, 95, 105, 114, 183, 437, 609, 682, 827, 902. These results were published in the On-Line Encyclopedia of Integer Sequences as OEIS A395686. Four further indices, 1090, 1153, 1215, 1378, were subsequently added to the sequence by Sean A. Irvine. For the Fibonacci sequence, starting from the cases already recorded in OEIS A382922, our search produced seven further Smith numbers: F1440, F1554, F1596, F1863, F2256, F2277, F2559. For the even-index cases, the identity F2n=FnLn allows available Fibonacci and Lucas factorization data to be combined to recover the complete factorizations needed for the Smith test. Surprisingly, the two searches also meet at index 827: both F827 and L827 are Smith numbers.
Related
- Two Primes and Five Mixed Fibonacci-Lucas Summands
- Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
- Monogenity of the irreducible factors of (k,t)-Fibonacci and (k,t)-Lucas polynomials
- The local Dold congruence for Fibonacci and Lucas sequences
- Cancellation profiles of Fibonacci-Lucas zeta sums
BibTeX
@misc{dario2026when,
title = {{When Fibonacci and Lucas Meet Smith: A Computational Exploration}},
author = {Ronie P. Dario and João L. Gonçalves and Moniky P. N. Oliveira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01955},
eprint = {2609.01955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01955},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - When Fibonacci and Lucas Meet Smith: A Computational Exploration
AU - Ronie P. Dario
AU - João L. Gonçalves
AU - Moniky P. N. Oliveira
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01955
UR - https://arxiv.org/abs/2609.01955
AB - The Fibonacci and Lucas sequences are old friends in recreational number theory. Here we ask a simple question: which of their terms F_n and L_n are Smith numbers? A Smith number is a composite integer whose decimal digit sum equals the sum of the decimal digits of its prime factors, counted with multiplicity. The question is easy to state, but for large n it quickly becomes a factorization problem. Using available complete factorization data, we carried out computational searches in both sequences and obtained new Smith terms in each of them. For the Lucas sequence, the first author's search found ten Smith terms with indices below 1000: 3, 95, 105, 114, 183, 437, 609, 682, 827, 902. These results were published in the On-Line Encyclopedia of Integer Sequences as OEIS A395686. Four further indices, 1090, 1153, 1215, 1378, were subsequently added to the sequence by Sean A. Irvine. For the Fibonacci sequence, starting from the cases already recorded in OEIS A382922, our search produced seven further Smith numbers: F_1440, F_1554, F_1596, F_1863, F_2256, F_2277, F_2559. For the even-index cases, the identity F_2n=F_nL_n allows available Fibonacci and Lucas factorization data to be combined to recover the complete factorizations needed for the Smith test. Surprisingly, the two searches also meet at index 827: both F_827 and L_827 are Smith numbers.
ER -