General Fibonacci Sequence whose Terms are all Composite/Mistake 2

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Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary


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.


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 ...