Determine all possible positive integers m such that:
√(4m - 7)
...........
√(m + 1)
is a rational number.
Provide adequate reasoning for your answer.
Lets start by rationalizing the denominator, then we have sqrt[(4m-7)*(m+1)]/(m+1).
Then for the fraction to evaluate to a rational number while m is a positive integer, we must have the radical then evaluate to an integer.
Thus we can say there is some positive x such that x^2 = (4m-7)*(m+1).
Then x^2 = 4m^2 - 3m - 7. By multiplying each side by 16 we can complete the square and stay within the integer domain: (4x)^2 = (8m-3)^2 + 121.
Move the (8m-3)^2 to the left side and factor as a difference of perfect squares: (4x+8m-3) * (4x-8m+3) = 121. For positive m the first factor is larger than the second factor. 121 only has two factorizations into positive integers: 121*1 and 11*11.
If we take 4x+8m-3=11 and 4x-8m+3=11 then m=7/4, but this is not an integer m and must be discarded.
If we take 4x+8m-3=121 and 4x-8m+3=1 then m=8 is a solution. Check: sqrt[4m-7]/sqrt[m+1] = sqrt[4*8-7]/sqrt[8+1] = sqrt[25]/sqrt[9] = 5/3, which is rational as required by the problem statement.
Solution
