New Prime Numbers

I have found some new prime numbers using the PROTH program of Yves Gallot.

This program is based on the following theorem:

**Proth Theorem (1878):**

Let N=k 2^{n} +1 where k
< 2^{n} . If there is an integer number a so that

a^{(n-1)/2} =-1(mod N)
therefore N is prime.

The Proth program is a test for primality of greater numbers defined as

k b^{n} + 1 or k b^{n}
- 1. The program is made to look for numbers of less than

5.000.000 digits and it is optimized for numbers of more than 1000 digits.

Using this Program, I have found the following prime numbers:

3239 2

^{12345}+ 1 with 3720 digits a=3, a=77551 2

^{12345}+ 1 with 3721 digits a=5, a=77595 2

^{12345}+ 1 with 3721 digits a=3, a=119363 2

^{12345}+ 1 with 3713 digits a=5, a=7

Since the exponents of the first three numbers are Smarandache numbers

Sm(5)=12345 we can call this type of prime numbers, prime numbers

of Smarandache.

Helped by the MATHEMATICA program, I have also found new prime numbers

which are a variant of prime numbers of Fermat. They are the following:

2^(2^n)·3^(2^n)-2^(2^n)-3^(2^n), for n=1,4,5,7.

It is important to mention that for n=7 the number which is obtained has 100 digits.

Chris Nash has verified the values n=8 to n=20, this last one being a number of

815.951 digits, obtaining that they are all composite. All of them have tiny factor

except n=13.

**REFERENCES:**

Smarandache Factors and Reverse Factors. Micha Fluren. Smarandache Notions

Journal Vol. 10. www.gallup.unm.edu/~Smarandache

The Prime Pages www.utm.edu/research/primes

**AUTHOR:**

* Sebastián Martín
Ruiz . *Avda. de Regla 43. Chipiona 11550 Spain