How much is enough?


If you notice a pattern, how many times do you have to check that it works before being certain. Six? Twenty? Two thousand?

Two favorite examples of mine that demonstrate that patterns can continue for a long time before going awry:

The first example is the polynomial f(n)=n2+41n+41. If you plug in any whole number for n from 1 to 40, then f(n) is a prime number: f(1)=83, f(2)=127, all the way to f(40)=3281. But f(41)=3403=41·83 is not prime. So something that works 40 times in a row might fail.

The second example are the cyclotomic polynomials. Look at polynomials of the form xn-1 that have been factored:


The last polynomial in each case is the cyclotomic polynomial of order n. [It has a much more technical definition using imaginary numbers and the product of primitive roots of unity]. And at first glance it looks like the coefficients are 0, 1, or -1. Even at second glance or sixth glance, since it’s true for the first 104 cyclotomic polynomials. But not the 105th.


See those two coefficients of 2 in that last polynomial? So the pattern of coefficients being only 0, 1, or -1 fails. Interestingly, the reason for this initial failure occurring so late in the game is that 105 is the smallest number that has three distinct odd prime factors (105=3·5·7). The integer 385 is also a product of three distinct odd primes (385=3·5·11) and the 385th cyclotomic polynomial is the first one to have a 3 as a coefficient (see Wolfram Mathworld).

Patterns. You just can’t trust them.


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