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 $F_n$ 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.