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)=n^{2}+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 x^{n}-1 that have been factored:

x-1=(x-1)

x^{2}-1=(x-1)(x+1)

x^{3}-1=(x-1)(x^{2}+x+1)

x^{4}-1=(x-1)(x+1)(x^{2}+1)

x^{5}-1=(x-1)(x^{4}+x^{3}+x^{2}+x+1)

x^{6}-1=(x-1)(x+1)(x^{2}+x+1)(x^{2}-x+1)

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 105^{th}.

x^{105}-1=(x-1)•(x^{2}+x+1)•(x^{4}+x^{3}+x^{2}+x+1)•

(x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1)•(x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1)•

(x^{12}-x^{11}+x^{9}-x^{8}+x^{6}-x^{4}+x^{3}-x+1)•

(x^{24}-x^{23}+x^{19}-x^{18}+x^{17}-x^{16}+x^{14}-x^{13}+x^{12}-x^{11}

+x^{10}-x^{8}+x^{7}-x^{6}+x^{5}-x+1)•

(x^{48}+x^{47}+x^{46}-x^{43}-x^{42}–2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}

+x^{34}+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}

+x^{15}+x^{14}+x^{13}+x^{12}-x^{9}-x^{8}–2x^{7}-x^{6}-x^{5}+x^{2}+x+1)

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 385^{th} 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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