> I don't think that Euler would have been fooled by something > as simple as x^2 + x + 41. After all, it is obvious that f(41) > is 41*43. He may have given this as an example of why not to > rely on anecdotal evidence. One of my favorites is the polynomial > (x-1)(x-2)(x-3)*...(x-n), multiplied out so that its form is not > so obvious. Then the prof/teacher says, f(1) = 0, f(2) = 0, ... > so therefore f(x) must always equal 0. Half the students are > convinced, and then the teacher takes f(n+1) = n!, which is > _very_ far from being 0.
A silly example: the "identity" (1+2+3+...+n)! = 1!3!5!...(2n-1)! which is true for n = 0, 1, 2, 3, and 4, but false for all larger values of n. Not in the same league with Euler's x^2+x+41 of course.
-- "Any clod can have the facts, but having opinions is an art."--McCabe