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 2n +1 where k < 2n . 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 bn + 1 or k bn - 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 212345 + 1 with 3720 digits a=3, a=7 7551 212345 + 1 with 3721 digits a=5, a=7 7595 212345 + 1 with 3721 digits a=3, a=11 9363 212345 + 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. http://fs.unm.edu/ The Prime Pages www.utm.edu/research/primes   AUTHOR: Sebastián Martín Ruiz . Avda. de Regla 43. Chipiona 11550 Spain