Existence of Poluet numbers in square form

In this paper, we will introduce theorem, which will generate Fermat Pseudo primes in different base system. Also, by fixing base-2, we find first poulet number and we show the first square type poulet number by suitable example and theorem.


INTRODUCTION
We know that, Primes are playing vital role in Computer Science, especially in Cryptography (see [1] & [2]) algorithms to strengthen security systems. In this connection, we are studying on Fermat Pseudo primes [3] (special primes) to more strengthen the cryptography in elegant way. As we know that, these Fermat Pseudo primes are born from Fermat little theorem and there is no much difference in both, except the prime case. Let us observe the definitions and generalizations of Fermat little theorem [4] and Fermat Pseudo primes. Or . We believe that, there are many such Pseudo primes are existing, and these can be classified by base system. Interestingly we have taken base-2 in the example-2 above. There are many Pseudo primes exist in different base system. At this point of time, we are interested in 2-base or bsae-2 ie., Thus, 341 is called as Fermat Pseudo prime as well as Poulet number. In the next section, we will discuss the generation of base-a Pseudo prime with theorem.

FERMAT PSEUDO PRIMES IN DIFFERENT BASE SYSTEM
This is the time to search number of availability of Fermat pseudo primes in different bases. Interestingly we came up with theorem, which generates infinitely many such primes. Let us state proposition and theorem. Proposition 2.1: For any integer a and prime p, the result of is an even integer. Proof: let us take a in two cases. Case#1: If a is odd integer, then we have Or , for some integer n. Case #2: If a is even integer, then we have Or , for some integer n.

The Bulletin of Society for Mathematical Services and Standards Vol. 10
Note: From the cited above theorem, one can generate many Fermat pseudo primes (not necessarily) in base-a. Let us plug some choose a and p values in the above theorem: Let us take a = 2 and p = 2. Since p does not divide a2 -1, we can generate pseudo prime m by the cited above theorem.
Here 5, is not pseudo prime as 5 does not have two equal or distinct prime factors. Let us take a = 2 and p = 3. Here p divides , we cannot generate pseudo prime m by the cited above theorem. Let us take a = 2 and p = 5. Here p does not divides , we can generate pseudo prime m by the cited above theorem.
Since we fixed base-2, one can call it is poulet number, instead of Fermat pseudo prime. In fact, 341 is first poulet number. Let us discuss the integer 121 is Fermat pseudo prime or poulet number. But Therefore, 121 is not Fermat pseudo prime as well as poulet number. But the following table says, there are some Fermat pseudo primes which are square numbers. Let us observe the following Table #1. for some The only possible m and n are 2 and 1 respectively. Since And from We get k = 3. Clearly, p must be 7. Similarly in other case, we get p = 3.

Proposition 2.4:
Find all primes p such that is perfect square. Proof: We are leaving the proof for readers. The above cited preposition 2.3 gave me an idea to search poulet numbers in square form, and interestingly we got the good stuff!