Primefree sequence

In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a₁ and a₂, such that the greatest common divisor ${\displaystyle \mathrm {gcd} (a_{1},a_{2})}$ is equal to 1, and such that for ${\displaystyle n>2}$ there are no primes in the sequence of numbers calculated from the formula

${\displaystyle a_{n}=a_{n-1}+a_{n-2}}$.

The first primefree sequence of this type was published by Ronald Graham in 1964.

A primefree sequence found by Herbert Wilf has initial terms

${\displaystyle a_{1}=20615674205555510,a_{2}=3794765361567513}$ (sequence A083216 in the OEIS)

The proof that every term of this sequence is composite relies on the periodicity of Fibonacci-like number sequences modulo the members of a finite set of primes. For each prime ${\displaystyle p}$, the positions in the sequence where the numbers are divisible by ${\displaystyle p}$ repeat in a periodic pattern, and different primes in the set have overlapping patterns that result in a covering set for the whole sequence.

The requirement that the initial terms of a primefree sequence be coprime is necessary for the question to be non-trivial. If the initial terms share a prime factor ${\displaystyle p}$ (e.g., set ${\displaystyle a_{1}=xp}$ and ${\displaystyle a_{2}=yp}$ for some ${\displaystyle x}$ and ${\displaystyle y}$ both greater than 1), due to the distributive property of multiplication ${\displaystyle a_{3}=(x+y)p}$ and more generally all subsequent values in the sequence will be multiples of ${\displaystyle p}$. In this case, all the numbers in the sequence will be composite, but for a trivial reason.

The order of the initial terms is also important. In Paul Hoffman's biography of Paul Erdős, The man who loved only numbers, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime ${\displaystyle 439351292910452432574786963588089477522344721}$.^[1]

Several other primefree sequences are known:

${\displaystyle a_{1}=331635635998274737472200656430763,a_{2}=1510028911088401971189590305498785}$ (sequence A083104 in the OEIS; Graham 1964),

${\displaystyle a_{1}=62638280004239857,a_{2}=49463435743205655}$ (sequence A083105 in the OEIS; Knuth 1990), and

${\displaystyle a_{1}=407389224418,a_{2}=76343678551}$ (sequence A082411 in the OEIS; Nicol 1999).

The sequence of this type with the smallest known initial terms has

${\displaystyle a_{1}=106276436867,a_{2}=35256392432}$ (sequence A221286 in the OEIS; Vsemirnov 2004).

• Graham, Ronald L. (1964). "A Fibonacci-like sequence of composite numbers" (PDF). Mathematics Magazine. 37 (5): 322–324. doi:10.2307/2689243. JSTOR 2689243.
• Knuth, Donald E. (1990). "A Fibonacci-like sequence of composite numbers". Mathematics Magazine. 63 (1): 21–25. doi:10.2307/2691504. JSTOR 2691504. MR 1042933.
• Wilf, Herbert S. (1990). "Letters to the Editor". Mathematics Magazine. 63: 284. doi:10.1080/0025570X.1990.11977539. JSTOR 2690956.
• Nicol, John W. (1999). "A Fibonacci-like sequence of composite numbers" (PDF). Electronic Journal of Combinatorics. 6 (1): 44. doi:10.37236/1476. MR 1728014.
• Vsemirnov, M. (2004). "A new Fibonacci-like sequence of composite numbers" (PDF). Journal of Integer Sequences. 7 (3): 04.3.7. Bibcode:2004JIntS...7...37V. MR 2110778.

• Problem 31. Fibonacci- all composites sequence. The prime puzzles and problems connection.
• "Primefree sequence". PlanetMath.
• Weisstein, Eric W. "Primefree Sequence". MathWorld.