A Pierpont prime is a prime number of the form: 2u3v + 1 for some non-negative integers  u  and  v .



A Pierpont prime of the second kind is a prime number of the form: 2u3v - 1 for some non-negative integers  u  and  v .


The term "Pierpont primes" is generally understood to mean the first definition, but will be called "Pierpont primes of the first kind" on this page to distinguish them.



Task

 Write a routine (function, procedure, whatever) to find Pierpont primes of the first & second kinds.
 Use the routine to find and display here, on this page, the first 50 Pierpont primes of the first kind.
 Use the routine to find and display here, on this page, the first 50 Pierpont primes of the second kind
 If your language supports large integers, find and display here, on this page, the 250th Pierpont prime of the first kind and the 250th Pierpont prime of the second kind.


See also

 Wikipedia - Pierpont primes
 OEIS:A005109 - Class 1 -, or Pierpont primes
 OEIS:A005105 - Class 1 +, or Pierpont primes of the second kind


import random
 
# Copied from https://rosettacode.org/wiki/Miller-Rabin_primality_test#Python
def is_Prime(n):
    """
    Miller-Rabin primality test.
 
    A return value of False means n is certainly not prime. A return value of
    True means n is very likely a prime.
    """
    if n!=int(n):
        return False
    n=int(n)
    #Miller-Rabin test for prime
    if n==0 or n==1 or n==4 or n==6 or n==8 or n==9:
        return False
 
    if n==2 or n==3 or n==5 or n==7:
        return True
    s = 0
    d = n-1
    while d%2==0:
        d>>=1
        s+=1
    assert(2**s * d == n-1)
 
    def trial_composite(a):
        if pow(a, d, n) == 1:
            return False
        for i in range(s):
            if pow(a, 2**i * d, n) == n-1:
                return False
        return True  
 
    for i in range(8):#number of trials 
        a = random.randrange(2, n)
        if trial_composite(a):
            return False
 
    return True
 
def pierpont(ulim, vlim, first):
    p = 0
    p2 = 1
    p3 = 1
    pp = []
    for v in xrange(vlim):
        for u in xrange(ulim):
            p = p2 * p3
            if first:
                p = p + 1
            else:
                p = p - 1
            if is_Prime(p):
                pp.append(p)
            p2 = p2 * 2
        p3 = p3 * 3
        p2 = 1
    pp.sort()
    return pp
 
def main():
    print "First 50 Pierpont primes of the first kind:"
    pp = pierpont(120, 80, True)
    for i in xrange(50):
        print "%8d " % pp[i],
        if (i - 9) % 10 == 0:
            print
    print "First 50 Pierpont primes of the second kind:"
    pp2 = pierpont(120, 80, False)
    for i in xrange(50):
        print "%8d " % pp2[i],
        if (i - 9) % 10 == 0:
            print
    print "250th Pierpont prime of the first kind:", pp[249]
    print "250th Pierpont prime of the second kind:", pp2[249]
 
main()