7x2-44x+12 = (7x-2)(x-6).
Because the expression can be factored. it can only have a prime value if (x-6) = 1, or if both terms are perfect powers of the same prime.
Assume the latter is true. Then (7x-2) = y^a, (x-6) = y^b.
The difference between the terms is (6x+4).
Since x^n-x^(n-1)=(x-1)x^(n-1), no two powers of x are closer than x and x^2, so say x^2-x = (6x+4), when x is between 7 and 8. For any larger powers, say x^4-x^3=(6x+4), x must be less than that. So x cannot exceed 7, and the assumption is easily shown to be false.
Consequently, (x-1)= 6, so x=7, when (7x-2) = 47, a prime number, and N=1.
Note 1: Excel gives 1087^2 as the solution when x=414.
Note 2: There is an infinite number of non-prime solutions to 7x2-44x+12 = y^2. The first few values of x are 6, 26, 366, 5786, 92166, 1468826,...
Edited on October 24, 2019, 10:11 pm
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Posted by broll
on 2019-10-24 21:58:27 |
Possible solution
