What is the most perplexing and difficult unsolved math problem?

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Discussion Overview

The discussion revolves around identifying the most perplexing and difficult unsolved math problem. Participants explore various conjectures and the nature of mathematical difficulty, touching on historical context and the evolution of mathematical thought.

Discussion Character

  • Debate/contested
  • Exploratory
  • Conceptual clarification

Main Points Raised

  • Some participants suggest the Extended Riemann Hypothesis (ERH) as a significant unsolved problem.
  • One participant highlights the Collatz conjecture as an example of a problem that appears simple but is notoriously difficult to solve, noting that many mathematicians have struggled with it for years.
  • Another participant reflects on historical geometric problems posed by the ancient Greeks, suggesting that the most difficult unsolved problem might be one that takes centuries to resolve.
  • There is mention of the continuum hypothesis, which deals with the cardinality of subsets of real numbers, as a particularly perplexing issue in set theory.
  • Participants express that the difficulty of unsolved problems is subjective and can vary greatly, with some problems appearing simple yet being complex to prove.

Areas of Agreement / Disagreement

Participants express a range of opinions on what constitutes the most difficult unsolved problem, with no consensus reached. Multiple competing views remain regarding the nature and examples of such problems.

Contextual Notes

Participants acknowledge that the difficulty of mathematical problems can be subjective and that historical context plays a role in understanding their complexity. There are references to problems that have eluded resolution for over a century, indicating a long-standing nature of some mathematical inquiries.

ElliotSmith
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TL;DR
What is the most perplexing and difficult unsolved math problem you can think of?
What is the most perplexing and difficult unsolved math problem that you can possibly think of?
 
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ERH.
 
We generally don't know how difficult a problem is until we solve it. We only set lower bounds on the difficulty. If many leading mathematicians can't solve it after years of work it's probably a hard problem.

The Collatz conjecture is a prominent example of a problem that looks simple but is very hard. You can explain the problem to a 5 year old - but despite many attempts we don't have a solution. Erdős said "Mathematics may not be ready for such problems."
 
fresh_42 said:
ERH.
what is this?
 
The ancient greeks wondered about geometric problems like whether it was possible to draw a square with the same area as a given circle, or whether it was possible to trisect an angle. Humanity didn't prove those to be impossible until the 1800s - it took over 2000 years to go from hypothesis to proof! Enormously important new fields of math were invented to solve these seemingly straightforward problems.

The most difficult unsolved math problem that we have right now is whichever one will take 2000 years to solve. We might never actually see something like that again, since the field of mathematics has matured so much, but there are still plenty of examples of questions that are 100+ years old that we still can't answer.
 
I can't imagine the continuum hypothesis which stated that we couldn't know whether there's a subset of R could have the cardinality exact bigger than N and excatly smaller than R. maybe I would recommend the set theory. it's really weird.
 
  • #10
Any problem that is unsolved is unsolved for a reason. It is hard to say it is difficult or easy because it is unsolved! There are some theorems that don't seem too bad, but are actually incredibly hard to prove and others that seemed awful, but ended up having relatively simple proofs. There's no way to really tell. Perhaps the most famous unsolved problem is the Collatz Conjecture because it is so easily stated and yet has eluded all of humanity perhaps since shortly after humans first learned to count.
 

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