5th order equation unsolvable by approximation methods?

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Discussion Overview

The discussion revolves around the equation x^5 - x + 1 = 0, focusing on its solvability through analytical and approximation methods. Participants explore the implications of Galois theory, the potential for singularities in approximation methods, and the relationship to Gödel's criteria for unsolvable equations.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation

Main Points Raised

  • Some participants assert that the equation x^5 - x + 1 = 0 has no known analytical solutions and question whether approximation methods can reliably yield solutions or if the equation is intrinsically unsolvable.
  • Others propose that approximation methods, such as binary search, can be applied effectively given the continuity of the polynomial and the existence of points with opposite signs.
  • A participant mentions that Galois theory indicates that polynomial equations of degree higher than four cannot be solved analytically, but this does not necessarily apply to all analytic techniques.
  • Another participant references a similar equation discussed by Julian Havil, noting that while some equations cannot be expressed in radicals, they still have real roots that can be approximated.
  • Some participants express a desire for a continuous function that is absolutely unsolvable by any method, but one participant concludes that continuous functions can be graphed and thus solved through various means.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of approximation methods to the equation, with some believing in their effectiveness while others remain skeptical. There is no consensus on whether the equation is intrinsically unsolvable or if it fits Gödel's criteria.

Contextual Notes

Participants reference various mathematical theories and concepts, including Galois theory and Gödel's criteria, but there are unresolved assumptions regarding the limitations of approximation methods and the definitions of solvability.

arupel
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I am across this equation with no known analytical solutions:

x^5 - x + 1 = 0.

I asked this before and the answer was that you can get as good an approximation you want by approximation methods.

It is possible that when applying approximation methods, you will get singularities. These methods won't work. The equation is bullet proof.

The question is can we be cetain that there are approximation methods that will work or is this equation intrincally unvolvable even by approximatiuon methods?

My second question is messy. It is a stray thought.
Is this equaton one that satisified Godel's criteria for an unsolveable equation is tue but can be shown to be not provable and not unproveable? If it fits Godel's criteria it would be nice to show studends.

PS: Anyway, In teaching algebra, it is nice to show this unsolveable simple equation.
 
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Binary search always works [on continuous functions defined over a contiguous range, given a pair of starting points]. If you can find a point where the graph of the function is positive and a point where the graph is negative then a binary search will converge toward a point where it is zero. By inspection, this polynomial is negative at x=-1000 and positive at x=+1000. Like all polynomials it is continuous and is defined over a contiguous range.

No, this has nothing to do with Godel.

Edit: Added some caveats to dot the i's and cross the t's
 
Starting with the interval ##[-2,-1]##, the solution is inside and you can approximate it with bisection method (for example) ... it seems to work ...
 
arupel said:
My second question is messy. It is a stray thought.
Is this equaton one that satisified Godel's criteria for an unsolveable equation is tue but can be shown to be not provable and not unproveable? If it fits Godel's criteria it would be nice to show studends.

If you want any example for Godel see Goodstein's[/PLAIN] theorem. Note that Goodstein can be expressed in PA, but not provable in PA. It is provable in ZFC.
 
Last edited by a moderator:
arupel said:
I am across this equation with no known analytical solutions:

x^5 - x + 1 = 0.

I asked this before and the answer was that you can get as good an approximation you want by approximation methods.

It is possible that when applying approximation methods, you will get singularities. These methods won't work. The equation is bullet proof.

The question is can we be cetain that there are approximation methods that will work or is this equation intrincally unvolvable even by approximatiuon methods?

My second question is messy. It is a stray thought.
Is this equaton one that satisified Godel's criteria for an unsolveable equation is tue but can be shown to be not provable and not unproveable? If it fits Godel's criteria it would be nice to show studends.

PS: Anyway, In teaching algebra, it is nice to show this unsolveable simple equation.
Galois proved that polynomial equations (in general) higher than 4th degree cannot be solved analytically.
 
mathman said:
Galois proved that polynomial equations (in general) higher than 4th degree cannot be solved analytically.

That's a really inaccurate statement of what has actually been proved.
First, it's Abel and Ruffini who proved it first, not Galois. Although Galois' proof is superior.
Second, they are unsolvable using radicals, not with other analytic techniques. There are in fact some analytic techniques which can solve all 5 degree equations.
Third, this thread is about approximation, for which Galois theory is completely irrelevant.
 
A similar equation is discussed by Julian Havil in his book ,The Irrationals- A Story of the Numbers You Can't Count On". The equation is
x^5- x- 1=0. Julian says "no finite expression formed of the composition of radicals will provide an explicit expression for its roots, with its only real root
equal to 1.1673039782614186843...". p.133
Also the innocent looking equation: x^5- 5x+12=0 requires about 600 symbols for its exact expression in radicals for the only one real solution -1.842085966... p.133
 
Thanks. I was hoping for a mystery; a continuous function which is absolutely unsolveble either analyticaly or by approximation methods, a platonic equation.

Sadly the mundane fact is that if it is a continious function it can be plotted. If it can be plotted it can be solved by the graph or analytically or by approximation methods, an obvious fact.

signing off
 

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