**MURARI PRIMES**

This is my own research work as a first year undergrad at Fergusson College,Pune.

I am writing this note in layman's language not as a research paper.

First, We take an Odd Prime Number and then we convert it to its Binary Equivalent. Then from that Binary equivalent we take first '1' and place it at last of Binary Equivalent. Then again we convert that Binary Equivalent to its respective decimal number.

Let us assume, P = 409 (Which is a prime number) and convert it to its binary equivalent.

We get, B = 110011001

Then , Take the first '1' from its 'B' and place it at the last of itself. So, New Binary number is

B' = 100110011

Now we convert B' to its decimal equivalent,

we get, D' = 307 ,Which is Prime.

If we use the above process for Odd Prime Numbers then we will get one of the following results :

1. The number D' will be a prime number.
2. D' will be an Arithmetic Mean of prime numbers just before and after that number.
3. Say, P = Prime number just before D'. Then D' = a +1
4. D' = P +/- 'a prime number'

The above Algorithm can be used for generating prime numbers.

The Prime Numbers which follow the listed properties, are being called as "Murari Primes" by me.

For D = 409, we get D' = 307(a prime number) . It follows property number 1.

For D = 23, we get D' = 15(Arithmetic Mean of 13 and 17) . It follows property number 2.

For D = 383, we get D' = 255((Arithmetic Mean of 251 and 257) + 1). It follows property number 3.

For D = 173, we get D' = 91((Arithmetic Mean of 89 and 97)-2). It follows property number 4.

Note -

• First 100 consecutive prime numbers are 'Murari Primes'
• This property is being followed by primes up to 7 digit Prime Numbers.
• Further research is going on 'Murari Primes'