New Directions for Primality Test

Abstract: In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method. It is conjectured that the algorithm finds a divisor of N in at most O(ln^{4} N), where ln denotes the logarithm with respect to base 2. The algorithm can be applied to find the next largest Mersenne prime number. Some directions are given regarding this. The second algorithm finds a prime divisor of N using the concept of graph pairs and it is proved that the time complexity of the second algorithm is at most O(ln^{2} N) for infinitely many cases (for approximately large N). The advantages and disadvantages of the second algorithm are also analyzed.