General Fibonacci Sequence whose Terms are all Composite/Mistake 2
Source Work
1986: David Wells: Curious and Interesting Numbers:
- The Dictionary
- $1,786,772,701,928,802,632,268,715,130,455,793$
1997: David Wells: Curious and Interesting Numbers (2nd ed.):
- The Dictionary
- $1,786,772,701,928,802,632,268,715,130,455,793$
Mistake
- Together with $1,059,683,225,053,915,111,058,165,141,686,996$, the start of a generalized Fibonacci sequence (in which each term is the sum of the previous two) in which every member is composite although the first $2$ terms have no common factor.
Correction
This pair was originally presented by Ronald Lewis Graham in a paper of $1964$.
However, in $1990$ Donald Ervin Knuth pointed out that:
- Incidentally the values ... are not the same as the $34$-digit values in Graham's original paper. A minor slip caused his original numbers to be respectively congruent to $F_{32}$ and $F_{33} \pmod {1087}$, not to $F_{33}$ and $F_{34}$, although all the other conditions were satisfied. Therefore the sequences defined by his published starting values may contain a prime number $A_{64 n + 31}$.
While the second edition of Curious and Interesting Numbers has added Knuth's new pair from his $1990$ paper ($62,638,280,004,239,857$ and $49,463,435,743,205,655$), Wells has neglected to remove the original paragraph that he published concerning $1,786,772,701,928,802,632,268,715,130,455,793$ and $1,059,683,225,053,915,111,058,165,141,686,996$.
Also note the tentative nature of Knuth's note:
- may contain a prime number $A_{64 n + 31}$.
So even at this point it is not certain that the two numbers presented in Graham's original paper do in fact fail to form a generalized Fibonacci sequence with these properties.
And even then, the mistake may have originated with John Brillhart. As Graham states in his $1964$ paper:
- I am grateful to Mr. John Brillhart for his assistance in obtaining an explicit solution to $(2)$. In particular ...
Sources
- Nov. 1964: R.L. Graham: A Fibonacci-Like Sequence of Composite Numbers (Math. Mag. Vol. 37, no. 5: pp. 322 – 324) www.jstor.org/stable/2689243
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,786,772,701,928,802,632,268,715,130,455,793$
- Feb. 1990: Donald E. Knuth: A Fibonacci-Like Sequence of Composite Numbers (Math. Mag. Vol. 63, no. 1: pp. 21 – 25) www.jstor.org/stable/2691504
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,786,772,701,928,802,632,268,715,130,455,793$