Primefree Sequence (Paperback)

High Quality Content by WIKIPEDIA articles In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it generally 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 GCD(a1, a2) = 1, and such that for n > 2 there are no primes in the sequence of numbers calculated from the formula: an = an 1 + an 2. 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 p, the positions in the sequence where the numbers are divisible by 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.