A Solution for Finding Composite Numbers in an Unending Sequence Starting with Prime Numbers

A non-terminating sequence like 31, 331, 3331, 33331, … starts with first seven terms as prime numbers, while the 8th term, which is 333333331, can be expressed as 17 x 19607843. Using Fermat’s Little Theorem, it can be easily proved that there are many more terms in this sequence that are not prime numbers. This paper puts forward a solution to find factors of composite numbers in all such sequences without using Fermat’s Little theorem or divisibility tests. The solution uses a prime number only once to scan all the terms of the unending sequence together, to check if any term is divisible by that prime number instead of checking every term separately, hence reduces the computational complexity. The solution finds the smallest number of the sequence which is divisible by a particular prime number and also proves that it cannot be assumed that all the terms of such sequences will be prime numbers.