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 Does It Continue? 11 (Posted on 2019-02-10)
Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Lets factor polynomials of the form x^n-1. Starting with 1 the following list can be generated:
n=1: (x-1)
n=2: (x-1)*(x+1)
n=3: (x-1)*(x^2+x+1)
n=4: (x-1)*(x+1)*(x^2+1)
n=5: (x-1)*(x^4+x^3+x^2+x+1)
n=6: (x-1)*(x+1)*(x^2+x+1)*(x^2-x+1)

One thing to notice is that each line has exactly one polynomial factor not seen earlier in the list:
n=1: x-1
n=2: x+1
n=3: x^2+x+1
n=4: x^2+1
n=5: x^4+x^3+x^2+x+1
n=6: x^2-x+1

Does each new factorization always produce exactly one new polynomial factor?

Looking more closely you may see that all the coefficients are -1, 0, or 1. Does this continue to be the case for all factors?

 No Solution Yet Submitted by Brian Smith Rating: 5.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: Answering a different question... | Comment 2 of 3 |

Actually, when n=105, there is 2 and -2 in the factors of x^105-1.

 Posted by Math Man on 2019-02-13 11:05:30

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