2. Special Quadratic Functions An Investigation

3. Special Quadratic Functions Consider the quadratic function  f(n) = n 2 + n + 41 If we put n = 0, we get f(0) = 0 2 + 0 + 41 = 41 If we put n = 1, we get f(1) = 1 2 + 1 + 41 = 43. Copy and complete the table below. What do you notice about the values of f(n)? What kind of special numbers are they? Is this type of number obtained for all values of n? Investigate. Try putting n = 10, 15, 20, 25, 30, 35, 40,- - etc- n0123456789 f(n)4143

4. Exercise Make up a similar table of values for the function f(n) = 2n 2 + 29. What type of number does this formula generate? Can you find a quadratic formula which generates these special numbers, one after the other until it fails?

5. Solution The function f(n) = n 2 + n + 41 generates prime numbers for n [0,39]. The numbers generated are called Euler numbers. Other quadratic prime generators include FunctionRangeAttributed to f(n) = 2n 2 +29[0,39]Legendre f(n) = n 2 +n + 17[0,15]Legendre f(n) = 36n 2 -810n + 2753 [0,44]Fung