Re: [math-fun] Two little sextics,
On Wed, Dec 28, 2016 at 5:52 PM, <rcs@xmission.com> wrote:
For the 2nd, the subst x->iy converts it to symmetric, and then y+1/y -> z gets it down to a cubic. I don't see anything for the 1st. Rich
Yeah, I just wrote up y+1/y -> z for a dangerous youngster named Gabe. gosper.org/gabesolve.nb . It's a Mathematica notebook, but there's a free reader. http://library.wolfram.com/infocenter/TechNotes/426/ Or at least there was. The link now redirects to their Computable Document player. Tell me if this won't view the .nb, and I'll write a cdf. --rwg
Nice roots, nice coefficients: In[414]:= Factor@Decompose[MinimalPolynomial[1/2 (-1 - 5^(1/3) + 5^(1/6) Sqrt[2 + 5^(1/3) + 5^(2/3)]), x], x] Out[414]= {-1 + 3 x + 3 x^5 + x^6} Nice roots, nasty coefficients: Out[1020]= (-3 I + Sqrt[3 + 4 2^(1/3) 3^(2/3)])^6 In[1021]:= Factor@Decompose[MinimalPolynomial[%, x], x] Out[1021]= {570630428688384000000 + 4891824455002619904 x + 161093791317491712 x^2 + 12153384861696 x^3 + 984379392 x^4 - 17280 x^5 + x^6} So the "parenthesand" has degree 36. --rwg
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Quoting Bill Gosper <billgosper@gmail.com>:
irreducible, indecomposable, nonreciprocal, solvable: In[996]:= Factor@Decompose[MinimalPolynomial[I/Sqrt[3]-(I (1/2 (9 I-2 Sqrt[3]+3 3^(1/4) Sqrt[-4 I-3 Sqrt[3]]))^(1/3))/3^(2/3)+(I+Sqrt[3])/(2^(2/3) (27 I-6 Sqrt[3]+9 3^(1/4) Sqrt[-4 I-3 Sqrt[3]])^(1/3)),x],x] Out[996]= {3-3 x+x^6}
In[997]:= Factor@Decompose[MinimalPolynomial[1/12 (2-2^(2/3) (-29+3 Sqrt[321])^(1/3)+2^(2/3) (29+3 Sqrt[321])^(1/3)-2^(5/6) Sqrt[3 (7 2^(1/3)+(83-3 Sqrt[321])^(1/3)+(83+3 Sqrt[321])^(1/3))]),x],x] Out[997]= {-1-x-x^5+x^6} --rwg
a = (-3 I + Sqrt[3 + 4 2^(1/3) 3^(2/3)]) is of degree 12. The minimum polynomial for a^2 is x^6 + 36 x^5 + 864 x^4 + 9792 x^3 + 456192 x^2 - 3234816 x + 8294400 and the minimum polynomial for a^2/4 is x^6 + 9 x^5 + 54 x^4 + 153 x^3 + 1782 x^2 - 3159 x + 2025 = \prod (x^2+(3-2*r)*x+(r+3)^2) r runs over all cube roots of 18. Warut On Thu, Dec 29, 2016 at 10:19 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Wed, Dec 28, 2016 at 5:52 PM, <rcs@xmission.com> wrote:
For the 2nd, the subst x->iy converts it to symmetric, and then y+1/y -> z gets it down to a cubic. I don't see anything for the 1st. Rich
Yeah, I just wrote up y+1/y -> z for a dangerous youngster named Gabe. gosper.org/gabesolve.nb . It's a Mathematica notebook, but there's a free reader. http://library.wolfram.com/infocenter/TechNotes/426/ Or at least there was. The link now redirects to their Computable Document player. Tell me if this won't view the .nb, and I'll write a cdf. --rwg
Nice roots, nice coefficients: In[414]:= Factor@Decompose[MinimalPolynomial[1/2 (-1 - 5^(1/3) + 5^(1/6) Sqrt[2 + 5^(1/3) + 5^(2/3)]), x], x]
Out[414]= {-1 + 3 x + 3 x^5 + x^6}
Nice roots, nasty coefficients: Out[1020]= (-3 I + Sqrt[3 + 4 2^(1/3) 3^(2/3)])^6
In[1021]:= Factor@Decompose[MinimalPolynomial[%, x], x]
Out[1021]= {570630428688384000000 + 4891824455002619904 x + 161093791317491712 x^2 + 12153384861696 x^3 + 984379392 x^4 - 17280 x^5 + x^6}
So the "parenthesand" has degree 36. --rwg
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Quoting Bill Gosper <billgosper@gmail.com>:
irreducible, indecomposable, nonreciprocal, solvable: In[996]:= Factor@Decompose[MinimalPolynomial[I/Sqrt[3]-(I (1/2 (9 I-2 Sqrt[3]+3 3^(1/4) Sqrt[-4 I-3 Sqrt[3]]))^(1/3))/3^(2/3)+(I+Sqrt[3])/(2^(2/3) (27 I-6 Sqrt[3]+9 3^(1/4) Sqrt[-4 I-3 Sqrt[3]])^(1/3)),x],x] Out[996]= {3-3 x+x^6}
In[997]:= Factor@Decompose[MinimalPolynomial[1/12 (2-2^(2/3) (-29+3 Sqrt[321])^(1/3)+2^(2/3) (29+3 Sqrt[321])^(1/3)-2^(5/6) Sqrt[3 (7 2^(1/3)+(83-3 Sqrt[321])^(1/3)+(83+3 Sqrt[321])^(1/3))]),x],x] Out[997]= {-1-x-x^5+x^6} --rwg
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participants (2)
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Bill Gosper -
Warut Roonguthai