  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.
