Can a math problem have a huge (or infinite) step-solution that's non-repetitive?

[Kyriakos]

Well, can it?

I can imagine problems that just require going on and on, but they usually repeat the same procedure, and even if they use some others they don't lead to any newer ones later on.
Intuitive i would suppose that no problem which is solvable can require huge numbers of steps to a solution, and thus that if a problem goes on in repeating steps it inevitably is not leading to more than an approximation.

Of course if new concepts are applied those problems may be solved, but again i suppose that would mean the solution is elegant and not spaced-out.

(and although it is not the sort of 'problem' i had in mind (i mean conjectures more or less), maybe it can be applied to pi as well, given that with current ways to look at it it won't even repeat even periodically, but not lead to a progress of its evaluation as a number either nomatter how many digits are presented).

 

[Agent327]

Probably, yes.

 

[Kyriakos]

^That is a brief reply, but it is not elegant if not also true at least in some facet (and the intended one at that as well)

 

[Agent327]

I'm making an educated guess.

Why don't you ask such questions on the Science forum?

 

[Kyriakos]

I'm making an educated guess.

Why don't you ask such questions on the Science forum?

Because i am against the counter-productive elitism of discussing scientific matters away from the general public.

Or it could be cause the science forum has next to no traffic

 

[Perfection]

http://en.m.wikipedia.org/wiki/Undecidable_problem

 

[Lord AJ]

Well there are differential equations of the 4th and higher orders that are solved by computers as they're to difficult and tedious for humans. So yes, there can be such long math problems.

 

[Kyriakos]

^But i suppose those follow steps that are similar to each other, even if they factor in some 'randomising' process? (eg in some other forum a poster briefly mentioned Lof randomness).

http://en.wikipedia.org/wiki/Algorithmically_random_sequence

But algorithms tend to have very similar steps (if not the iteration of just one step or two). I am asking about large numbers of not that tied to each other steps, eg requiring combinations between math orders and so on. Otherwise even the Sieve of Eratosthenes goes on forever, but after a while even a super-computer won't readily show results for the massive primes.

 

[Lord AJ]

^But i suppose those follow steps that are similar to each other, even if they factor in some 'randomising' process? (eg in some other forum a poster briefly mentioned Lof randomness).

http://en.wikipedia.org/wiki/Algorithmically_random_sequence

But algorithms tend to have very similar steps (if not the iteration of just one step or two). I am asking about large numbers of not that tied to each other steps, eg requiring combinations between math orders and so on. Otherwise even the Sieve of Eratosthenes goes on forever, but after a while even a super-computer won't readily show results for the massive primes.

You asked for a problem that has an infinite (kind of) number of steps. Well these differential equations are one hell of a thing to be solving. I am studying this topic (although the scope of study is limited to 2nd order differential equations ) still I researched a bit and found the info I gave above. Even if the algorithms programmed to solve these equations instruct the Computer to repetitively do the same thing, they still involve innumerable steps.

 

[Lord AJ]

Also I just came up with another such problem :
Consider an equation in x with the powers of x ranging from 99 to 1, including fractional powers. You would agree this will give anyone a really hard time. [emoji6] [emoji6] [emoji6] [emoji6] [emoji6]

 

[Kyriakos]

^Very interesting, thanks

 

[Borachio]

You asked for a problem that has an infinite (kind of) number of steps. Well these differential equations are one hell of a thing to be solving. I am studying this topic (although the scope of study is limited to 2nd order differential equations ) still I researched a bit and found the info I gave above. Even if the algorithms programmed to solve these equations instruct the Computer to repetitively do the same thing, they still involve innumerable steps.

2nd order partial differential equations! Gah! Gah!

 

[nc-1701]

I think it would be very easy to create hugely long problems, perhaps trivial depending on how strict we are about "using the same procedure". As far as a problem that can only be solved in infinite steps, that seems somewhat improbable to me. After all how would we differentiate such a problem from one that can't be solved? It doesn't seem impossible, but my intuition leans toward no.

 

[Leoreth]

Well, can it?

I can imagine problems that just require going on and on, but they usually repeat the same procedure, and even if they use some others they don't lead to any newer ones later on.
Intuitive i would suppose that no problem which is solvable can require huge numbers of steps to a solution, and thus that if a problem goes on in repeating steps it inevitably is not leading to more than an approximation.

Of course if new concepts are applied those problems may be solved, but again i suppose that would mean the solution is elegant and not spaced-out.

(and although it is not the sort of 'problem' i had in mind (i mean conjectures more or less), maybe it can be applied to pi as well, given that with current ways to look at it it won't even repeat even periodically, but not lead to a progress of its evaluation as a number either nomatter how many digits are presented).

Depends on your definition of "step" and "repetition" really. Is "do something random" a unique step every time?

It really depends on your definitions if this question can even be answered. If the answer is yes, I can only see it being proven by example. However, it's impossible to give an example of an explicit algorithm with infinite steps. If I define some rule that generates steps, are the steps truly non-repetitive, or just similar steps with different parameters?

So under these constraints I would say the question is unanswerable.

I'm not even sure if we can call something with infinite steps an algorithm in the first place though.

Aren't Latin letters just as, if not more common in math? It's solve for "ex", not solve for "chi".

 

[Kyriakos]

Depends on your definition of "step" and "repetition" really. Is "do something random" a unique step every time?

It really depends on your definitions if this question can even be answered. If the answer is yes, I can only see it being proven by example. However, it's impossible to give an example of an explicit algorithm with infinite steps. If I define some rule that generates steps, are the steps truly non-repetitive, or just similar steps with different parameters?

So under these constraints I would say the question is unanswerable.

I'm not even sure if we can call something with infinite steps an algorithm in the first place though.


Aren't Latin letters just as, if not more common in math? It's solve for "ex", not solve for "chi".

What i had in mind as non-repetitive would be something generating different types of step every now and then, and those steps (likely) applied to anything predating them so as to exponentially increase the complexity of the progress to a solution.
As for what type of difference, well, it would have to be obvious difference, and to use a non math example there is 'obvious' difference between generating letters and generating words (despite letters used again), but likely later on they can be tied in some meta-way, where they link back to non-language human sense/mental life.

So the procedure would go on and on, without any end to new steps, much like there is (very different phenomenon, but not in this parallelism) no end to new primes as long as the progression of positive integers goes on.

Crucial difference here being that the ever generating new phenomena would be types of steps to a solution, not actual numbers or parameters themselves.

Intuitively i doubt it can ever be so.

And *maybe* this also would mean that there aren't infinite such different steps in any expanded math either, ie not in a billion years, etc.