Determine all possible value(s) of a positive prime constant P such that the equation: 2^X + 3P = Y² has precisely two distinct solutions in (X, Y), where each of X and Y is a positive integer.

After creating a list of prime numbers under 10000, the first 28 powers of 2, and the first 15000 perfect squares, I ran the following python code, and came up with one solution.

P=11 ... (X, Y) = (4, 7) and (8, 17)

2^4 + 3*11 = 16 + 33 = 49 = 7^2
2^8 + 3*11 = 256 + 33 = 289 = 17^2

for p in primes:
    total=0
    values=[]
    for x2 in powers_of_two:
        y2=x2+3*p
        if squares.count(y2):
            total+=1
            values.append([powers_of_two.index(x2)+1,squares.index(y2)+1])
    if total==2:
        print(p,values)

Posted by Justin on 2010-01-24 16:12:14