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seen Jul 15 at 21:57

I'm interested in factoring numbers of the form $(a^n±b^n)/(a±b)$


May
18
comment Splitting polynomials
@Jyrki: Yes, and lhf seems to be suggesting that too.
May
18
comment Splitting polynomials
@lhf: My original question was motivated by Paul Garrett's paper, which is why I'm looking for a different sort of solution. I admit that what I'm looking for may be unobtainable, but I figure there's no harm in looking.
May
18
comment Splitting polynomials
I think I could prove it if I put my mind to it. Howeveer, I'm not too bothered, as this is not the solution I'm looking for. (Maybe what I'm looking for is unobtainable). Your second comment here mirrors what's in Paul Garrett's paper. See my comments to the next solution.
May
18
awarded  Self-Learner
May
18
awarded  Teacher
May
18
answered Splitting polynomials
May
18
comment Splitting polynomials
It's not the sort of solution I was expecting. I was hoping to see something along the lines of extracting square roots by long division (yes, I know this is not a square root, but there are some things in common with square roots), and this solution doesn't do that.
May
17
comment Splitting polynomials
Another example, for $p=11$ is$$161051 + 161051x + 73205x^2 + 14641x^3 -1331x^4 -1331x^5 -121x^6 + 121x^7 + 55x^8 + 11x^9 + x^{10}$$ together with its companion where the signs of the coefficients of the odd powers of x are reversed.
May
17
comment Splitting polynomials
@Jyrki: Thanks for all your help, it is most appreciated.
May
17
comment Splitting polynomials
Oh my gosh; I seem to have found some sort of solution. It involves Lucas's formula for Cyclotomic Polynomials (see Riesel, Table 24).
May
17
comment Splitting polynomials
Sorry, @Gerry, I should have mentioned: yes it is all related to Aurifeuillian factorization and cyclotomic polynomials.
May
17
comment Splitting polynomials
Thanks; this is very helpful. I assume that the product should go from $k=1$ and not $j=1$. I am definitely looking for a closed formula for the coefficients.
May
17
revised Splitting polynomials
got the example to line up properly
May
17
revised Splitting polynomials
added 2 characters in body
May
17
awarded  Student
May
17
revised Splitting polynomials
added an example
May
17
revised Splitting polynomials
typo in title
May
17
awarded  Editor
May
17
revised Splitting polynomials
got the sums to display properly at last!
May
17
asked Splitting polynomials