Interpolation and Integers
Let be interpolated by a quadratic polynomial . That is, and . For how many triples of does have integer (including zero) coefficients?
The answer is 32.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Suppose f is d x^2 + e x + f. Then this is a system of equations: d + e + f = a, 4 d + 2 e + f = b, 9 d + 3 e + f = c. Then the matrix representing the left side is {{1, 1, 1}, {4, 2, 1}, {9, 3, 1}}. Taking the inverse of this gives us {{1/2, -1, 1/2}, {-(5/2), 4, -(3/2)}, {3, -3, 1}}. Multiplying by the vector {a,b,c}^T, this gives {{a/2 - b + c/2}, {-((5 a)/2) + 4 b - (3 c)/2}, {3 a - 3 b + c}}. In order for this vector (coefficients of f) to have integer coefficients, a and c must have the same parity. So the total number of combinations of a,b,c is (2 2+2 2)*4 = 32. This is because b can be 4 numbers, and a and c can each be even in two ways and odd in two ways.