Transcript Special Quadratic Functions

Special Quadratic Functions
An Investigation
Special Quadratic Functions
Consider the quadratic function 
f(n) = n2 + n + 41
If we put n = 0, we get f(0) = 02 + 0 + 41 = 41
If we put n = 1, we get f(1) = 12 + 1 + 41 = 43.
Copy and complete the table below.
n
0
1
f(n)
41
43
2
3
4
5
6
7
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-
8
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Exercise Make up a similar table of
values for the function f(n) = 2n2 + 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?
Solution
The function f(n) = n2 + n + 41 generates prime
numbers for n  [0,39].
The numbers generated are called Euler numbers.
Other quadratic prime generators include
Function
Range
Attributed to
f(n) = 2n2+29
[0,39]
Legendre
f(n) = n2 +n + 17
[0,15]
Legendre
f(n) = 36n2 -810n +
2753
[0,44]
Fung