primefree sequence
Consider the sequence^{} defined by ${a}_{1}=20615674205555510$, ${a}_{2}=3794765361567513$ and ${a}_{n}={a}_{n-1}+{a}_{n-2}$ for all $n>2$. As it has been verified not to contain any primes, it is called a primefree sequence^{}. The initial terms must be coprime^{}, or else the lack of primes is a trivial consequence of the initial terms sharing a divisor^{} other than 1.
Any Fibonacci-like sequence will naturally exhibit some patterns in the factorizations of its terms in relation^{} to their indices. The initial terms are chosen so that these patterns cover any possible value of $n$. So, for our example sequence, discovered by Wilf in 1990, $2|{a}_{3x+1}$, $3|{a}_{4x+2}$, $5|{a}_{5x+1}$, $7|{a}_{8x}$, etc. for a finite number of potential prime factors^{} (and $x\ge 0$ in each case).
Order is very important: switching the initial terms can cause primes to arise in the sequence. Switching the initial terms in our example causes ${a}_{138}$ and a few others afterwards to be prime.
The example sequence is listed in A083216 of the OEIS.
References
- 1 P. Hoffman. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998.
- 2 H. Nicol. A Fibonacci-like sequence of composite numbers^{}. Electronic J. of Combinatorics 6, 1999.
- 3 H. S. Wilf. Letters to the Editor. Math. Mag. 63, 284, 1990.
Title | primefree sequence |
---|---|
Canonical name | PrimefreeSequence |
Date of creation | 2013-03-22 15:54:49 |
Last modified on | 2013-03-22 15:54:49 |
Owner | CompositeFan (12809) |
Last modified by | CompositeFan (12809) |
Numerical id | 7 |
Author | CompositeFan (12809) |
Entry type | Definition |
Classification | msc 11B39 |