[math-fun] The Solvable Octic x^8-x^7+29x^2+29 = 0

24 Aug 2011

      Really From: tpiezas@gmail.com

Hello all,

The octic,

x^8-x^7+29x^2+29 = 0   (eq.1)

(by Igor Schein) is solvable, but not as easy as merely factoring over a
square root extension. Rather, this can be solvable by the 29th root of
unity.

My solution to (eq.1) is,
[tweaked (with permission) by rwg on the basis of numerical evidence:
{8 x} =
{1-a-b+c+d-e-f-g,
 1+a+b+c-d-e-f+g,
 1-a+b-c-d+e-f-g,
 1+a-b-c+d+e-f+g,
 1-a-b-c-d-e+f+g,
 1+a+b-c+d-e+f-g,
 1+a-b+c-d+e+f-g,
 1-a+b+c+d+e+f+g}
]
where each {a,b,c,d,e,f,g} = Sqrt[4v_i+1] and the v_i are the 7 roots of the
septic,

8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0    (eq.2)

The solution of which was given by Peter Montgomery as,

v_i = 2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5+w^24+w^27)+
(w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+w^20+w^21)
and one can set w_i = {t, t^7, t^23, t^25, t^16, t^20, t^24}, and t =
exp(2Pi*I/29).
P.S. Similarly, a solvable 32-deg equation in x can be solved by a 31-deg
Lagrange resolvent in z in the form,

x = z1^(1/2) +/- z2^(1/2) +/- ... +/- z31^(1/2)

though, unfortunately, no explicit examples are yet known.

- Tito Piezas III
http://sites.google.com/site/tpiezas/
(Check this out! --rwg)

[rwg: the septic roots are all real:

v->-2 (2 cos((2 π)/29)+2 cos((3 π)/29)+cos((4 π)/29)-2 cos((5 π)/29)-cos((6
π)/29)+2 cos((7 π)/29)-sin(π/58)-sin((3 π)/58)-sin((7 π)/58)+sin((9
π)/58)+sin((11 π)/58)-sin((13 π)/58))
v->2 (cos(π/29)+cos((2 π)/29)-cos((4 π)/29)-cos((5 π)/29)-2 cos((6 π)/29)-2
sin(π/58)-sin((3 π)/58)-sin((5 π)/58)-sin((7 π)/58)-sin((9 π)/58)-2 sin((11
π)/58)+2 sin((13 π)/58))
v->2 (2 cos(π/29)-cos((2 π)/29)-cos((3 π)/29)+cos((4 π)/29)+cos((5
π)/29)-cos((7 π)/29)-2 sin((3 π)/58)-2 sin((5 π)/58)-2 sin((7 π)/58)+sin((9
π)/58)+sin((11 π)/58)-sin((13 π)/58))
v->-2 (2 cos(π/29)-cos((2 π)/29)+cos((3 π)/29)+cos((5 π)/29)+cos((6
π)/29)+cos((7 π)/29)+sin(π/58)-sin((3 π)/58)-2 sin((5 π)/58)-sin((7 π)/58)-2
sin((11 π)/58)+2 sin((13 π)/58))
v->-2 (cos(π/29)+cos((2 π)/29)-2 cos((3 π)/29)-2 cos((4 π)/29)-cos((5
π)/29)+cos((6 π)/29)-2 cos((7 π)/29)+sin(π/58)-sin((5 π)/58)-2 sin((9
π)/58)+sin((11 π)/58)-sin((13 π)/58))
v->2 (cos(π/29)+2 cos((2 π)/29)+cos((3 π)/29)+cos((4 π)/29)-2 cos((5
π)/29)+cos((6 π)/29)+cos((7 π)/29)+sin(π/58)+2 sin((3 π)/58)-sin((5 π)/58)+2
sin((7 π)/58)+sin((9 π)/58))
v->-2 (cos(π/29)-cos((3 π)/29)+2 cos((4 π)/29)-2 cos((6 π)/29)-cos((7
π)/29)-2 sin(π/58)+sin((3 π)/58)-sin((5 π)/58)+sin((7 π)/58)+2 sin((9
π)/58)-sin((11 π)/58)+sin((13 π)/58))]

Bill Gosper

Bill Gosper

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