Geodesic
A Fibonacci-like sequence of composite numbers
The electronic journal of combinatorics, Tome 6 (1999)
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In 1964, Ronald Graham proved that there exist relatively prime natural numbers $a$ and $b$ such that the sequence $\{A_n\}$ defined by $$ {A}_{n} =A_{n-1}+A_{n-2}\qquad (n\ge 2;A_0=a,A_1=b)$$ contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth's computation, Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham's construction and generalize Wilf's note, and show that the 12-digit pair $$(a,b)= (407389224418,76343678551)$$ also defines such a sequence.
DOI : 10.37236/1476
Classification : 11B39
Mots-clés : regular covering of integers 
@article{10_37236_1476,
     author = {John W. Nicol},
     title = {A {Fibonacci-like} sequence of composite numbers},
     journal = {The electronic journal of combinatorics},
     year = {1999},
     volume = {6},
     doi = {10.37236/1476},
     zbl = {0977.11008},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1476/}
}
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John W. Nicol. A Fibonacci-like sequence of composite numbers. The electronic journal of combinatorics, Tome 6 (1999). doi: 10.37236/1476

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