Finding Possible Values of n for Perfect Squares
Let n be a positive integer such that both 2n+12n+1 and 3n+13n+1 are perfect squares. Find all possible values of n.
Finding Possible Values of n for Perfect Squares
To find the possible values of n such that both 2n+12n+1 and 3n+13n+1 are perfect squares, we can approach the problem by setting up equations and solving for n.
Let's denote the perfect squares as m^2 and k^2, where m and k are integers.
Equation 1:
2n + 12n + 1 = m^2
Simplify this to:
14n + 1 = m^2
Equation 2:
3n + 13n + 1 = k^2
Simplify this to:
16n + 1 = k^2
Now, we have two equations:
1. 14n + 1 = m^2
2. 16n + 1 = k^2
To find the possible values of n, we need to solve these two equations simultaneously. Let's proceed with the calculations.
Solving for n:
From Equation 1:
14n = m^2 - 1
14n = (m - 1)(m + 1)
From Equation 2:
16n = k^2 - 1
16n = (k - 1)(k + 1)
Now, we need to find common factors for n in both equations. Since n is a positive integer, we look for values that satisfy the conditions.
One possible solution is when:
m - 1 = 2
m + 1 = 7
This gives us m = 3, which implies n = 1.
Therefore, the only possible value of n that satisfies both conditions is n = 1.
In conclusion, the only positive integer n that makes both expressions 2n+12n+1 and 3n+13n+1 perfect squares is n = 1.