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From Euclid to Spigotry
Assignment 19 (outline 15) 
Last modified: September 15, 2004
1a. Write Niven's function fn(x) for the case where n = 2, a=3 , b=1 as a polynomial in x.

b. Calculate fn"(x), fn""(x), and Fn(x), and confirm that your Fn(x) satisfies the equation
 Fn"(x) + Fn(x) = f(x). Even in this simple case, you may want to use Mathematica to do the algebra.

c. Confirm that F(0) and F(3) are both integers and that they are equal.

2a. Using Mathematica, find the smallest positive value of n for which the value of the function
h(n) =  is less than 1.  Then add 100 to this value and evaluate h again.
In Mathematica, n! is Factorial[n], and you can use h[n_] := (some formula)
to define the function.  Finding the right value of n requires some trial and error.

b. The proof done in class that Niven's integral can be made less than 1 suggests, for this special case, starting with B = h(200) and then, if it is greater than 1, adding enough to n to kill off all the powers of 2.  Do this using Mathematica. Recall that if B = 2m then m = log2B = log B/ log 2, where in the last step the logarithms can be taken to any base.