Foundations, logic and sets
Unsolved problems
An unsolved problem is a precise mathematical claim or question that has resisted every accepted proof and disproof, regardless of how persuasive computation may appear.
In one sentence
What is Unsolved problems?
An unsolved problem is a precise mathematical claim or question that has resisted every accepted proof and disproof, regardless of how persuasive computation may appear.
One idea, three depths
Understand Unsolved problems at your level
Build the intuition
It is a math question nobody has yet proved right or wrong.
Use the mathematics
Testing millions of examples can strengthen a conjecture, but one hidden counterexample can still defeat it; a proof must cover every permitted case.
Make it precise
Open problems expose the boundary between empirical evidence and deductive certainty. Their status depends on exact formulation, accepted axioms, peer verification and sometimes independence from an axiom system.
Try the idea
Interactive concept laboratory
27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 137 → 412 → 206 → 103 → 310 → 155 → 466 → 233 → 700 → 350 → 175 → … → 1
Important: this experiment is evidence for one starting value, not a proof for every positive integer.
Where this fits
When should you learn Unsolved problems?
The first encounter is not the final level. Many ideas begin visually, become computational in high school and become formal in college.
- 1Pre-high schoolPrepare foundations
- 2High schoolUsually introduced
- 3CollegeGreater depth
- 4AdvancedGreater depth
- Number theory
- Geometry
- Analysis
- Mathematical physics
- Research methods
- Curiosity
- Examples and counterexamples
- Logic and proof
- Experimentation
- Conjecture
- Research literacy
- Advanced mathematics
Not a prerequisite, but it teaches the difference between evidence, conjecture and proof.
Open the Zero-to-QM pathBuild the vocabulary
Six core ideas
a precise unproved claimUsually motivated by patterns, computation, analogy or partial theorems.
verified casesBuilds confidence and guides research but cannot replace a universal argument.
∃x: claim failsOne valid case can disprove a universal conjecture immediately.
theorem under conditionsMay solve special cases, establish bounds or show what any eventual solution must satisfy.
neither P nor ¬P followsSome statements cannot be decided from a selected axiom system, assuming it is consistent.
checkable proofA claimed solution becomes established only after specialists can inspect and validate every essential step.
From question to conclusion
Worked reasoning
Repeat n/2 when even and 3n+1 when odd. Must every positive start reach 1?
Computation says yes for enormous ranges, but no general proof is known.
Is every even integer greater than 2 a sum of two primes?
Extensive verification and partial theorems support it, but the full binary conjecture remains open.
Do all nontrivial zeta zeros have real part 1/2?
Many zeros have been checked and deep consequences are known, but a proof or counterexample remains absent.
Why it matters
Connections across mathematics and beyond
Searches cases, discovers patterns and tests consequences while leaving the universal proof problem intact.
Open problems make the distinction between confidence and logical necessity impossible to ignore.
Problems can remain open for centuries, be solved unexpectedly or be shown independent of accepted axioms.
Reading claim status carefully prevents ‘verified many times’ from being mistaken for ‘proved.’
Questions worth keeping
The surprising edge
Simple statements can be brutally hard
Collatz is understandable with elementary arithmetic but resists current methods.
A computer check can be a proof—within limits
Finite exhaustive verification can prove a finite claim if the program, scope and computation are themselves rigorously certified.
Unsolved does not mean nobody tried
Major open problems often sit atop vast literatures of equivalent forms, partial results and failed approaches.
Clear answers
Questions about Unsolved problems
What is Unsolved problems in simple terms?
It is a math question nobody has yet proved right or wrong.
When should students learn Unsolved problems?
Unsolved problems is usually introduced at the high school level and revisited with greater depth later. Typical subjects include Number theory, Geometry, Analysis, Mathematical physics, Research methods.
Why is Unsolved problems important?
An unsolved problem is a precise mathematical claim or question that has resisted every accepted proof and disproof, regardless of how persuasive computation may appear.
What should I know before studying Unsolved problems?
Useful prerequisites are Curiosity, Examples and counterexamples, Logic and proof.
Is Unsolved problems needed for quantum mathematics?
Optional: Not a prerequisite, but it teaches the difference between evidence, conjecture and proof.