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Primefree Sequence

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A primefree sequence is sequence whose terms are never prime. Graham (1964) proved that there exist relatively prime positive integers a and b such that the recurrence equation

a_n=a_(n-1)+a_(n-2)
(1)

with a_0=a and a_1=b contains no prime numbers.

In addition, Graham (1964) constructed a pair of numbers (one 33 digits and the other 34)

a=331635635998274737472200656430763
(2)
b=1510028911088401971189590305498785
(3)

satisfying this condition. Knuth (1990) subsequently found a 17-digit pair

a=62638280004239857
(4)
b=49463435743205655
(5)

satisfying the same conditions. Almost immediately, Wilf (1990) found a smaller pair (one 17 digits and the other 16)

a=20615674205555510
(6)
b=3794765361567513.
(7)

Note that Hoffman (1998, p. 159) inadvertently inverted the order of the Wilf (1990) pair, thus obtaining a sequence that has prime terms for n=138, 163, 190, 523, 1855, 3228, 3579, 6468, 7170, 10230, 12783, 17259, 60139, 91315, 97923, 101823, 156075, 182220, ... (OEIS A108156), with no others for n<=194202 (E. W. Weisstein, May 5, 2006).

Nicol (1999) subsequently found the 12-digit pair (a,b)=(407389224418,76343678551).

See also

Integer Sequence Primes, Linear Recurrence Equation, Prime Number

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References

Graham, R. L. "A Fibonacci-Like Sequence of Composite Numbers." Math. Mag. 37, 322-324, 1964.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 11 and 252, 1994.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.Knuth, D. E. "A Fibonacci-Like Sequence of Composite Numbers." Math. Mag. 63, 21-25, 1990.Nicol, J. W. "A Fibonacci-Like Sequence of Composite Numbers." Elec. J. Combin. 6, R44, 1-6, 1999.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 178, 1991.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 367, 1996.Rivera, C. "Problem 31. Fibonacci-All Composites Sequence." http://www.primepuzzles.net/problems/prob_031.htm.Sloane, N. J. A. Sequence A108156 in "The On-Line Encyclopedia of Integer Sequences."Wilf, H. S. Letters to the Editor. Math. Mag. 63, 284, 1990.

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Primefree Sequence

Cite this as:

Weisstein, Eric W. "Primefree Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimefreeSequence.html

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