- Many famous unsolved math problems can be stated in grade-school language.
- Yitang Zhang proved prime pairs exist within a gap of less than 70 million.
- Terence Tao showed the Collatz conjecture holds for almost all numbers.
Yitang Zhang was working at a Subway sandwich shop when he solved an unsolved math problem that had stumped number theorists for centuries.
The problem itself sounds almost trivial. Are there infinitely many pairs of prime numbers that sit just two apart? Primes like 11 and 13, or 599 and 601?
That question, the twin prime conjecture, is one of roughly a dozen unsolved math problems that share a maddening quality: a child can understand the question, but no mathematician alive can answer it.
Key figure
246
Current smallest proven prime gap, down from 70 million in 2013
When Simple Questions Become Impossible
The Collatz conjecture may be the purest example. Pick any positive number. If it is even, halve it. If it is odd, triple it and add one. Repeat.
Every number anyone has ever tested eventually spirals down to one. Computers have verified this for all integers up to 2.36 x 1021.
Yet no one can prove it always works.
Terence Tao, a Fields Medal laureate at UCLA, made the strongest progress in decades on the Collatz conjecture in 2019. His proof showed the conjecture holds for "almost all" numbers in a precise mathematical sense.
But Tao himself acknowledged his methods probably cannot yield a complete solution.
Jeffrey Lagarias, a University of Michigan mathematician and leading expert on the problem, has called it "a really dangerous problem" because researchers become obsessed with it.
What is a conjecture?
A conjecture is a statement mathematicians believe to be true based on evidence and pattern, but which lacks a formal proof. Until proven, even the most compelling conjecture remains officially uncertain.
The Sandwich Shop Mathematician Who Stunned Number Theory
The twin prime conjecture offered a similar kind of frustration for decades. Then Zhang, a lecturer at the University of New Hampshire who had published nothing since 2001, submitted a paper to the Annals of Mathematics in April 2013.
I regard it as a completely certain theorem, although I cannot prove it.
Leonhard Euler, on the Goldbach conjecture, 1742
Zhang proved that infinitely many pairs of primes exist within a gap of less than 70 million. The number was enormous, but the principle was historic.
Within a year, a collaborative effort led by Tao and independently by James Maynard of Oxford shrank that bound to 246. Under certain technical assumptions, mathematicians believe it could reach six. The final step to two remains elusive.
Peter Sarnak at the Institute for Advanced Study described Zhang's work simply: "His result was spectacular."
Why the Easiest Questions Resist the Hardest Minds
Goldbach's conjecture joins the list. Every even number greater than two is the sum of two primes. Leonhard Euler believed it in 1742. Computers have confirmed it for staggeringly large numbers. Proof, after nearly three centuries, remains absent.
The Riemann hypothesis sits at the summit. It concerns the distribution of prime numbers through a function proposed by Bernhard Riemann in 1859. It carries a million-dollar Millennium Prize and, if solved, would unlock cascading progress across number theory.
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→Six of the seven Millennium Prize Problems remain open. Only the Poincare conjecture has fallen, solved by Grigori Perelman in 2003.
The pattern is consistent: questions about prime numbers, simple sequences, and basic arithmetic resist formal proof with a stubbornness that belies their apparent simplicity.
Zhang, now a professor at Sun Yat-sen University, continues working on number theory. The gap between 246 and 2 remains open.
Sources
- Primary Source: Every Unsolved Math Problem That Sounds Easy (ThoughtThrill, YouTube)
- Additional Context:
- Unheralded Mathematician Bridges the Prime Gap (Quanta Magazine)
- Mathematician Proves Huge Result on 'Dangerous' Problem (Quanta Magazine)
- Yitang Zhang's Spectacular Mathematical Journey (Institute for Advanced Study)
Fact Check: Claim-by-Claim Verification Verified
The article accurately represents all major claims about unsolved math problems, including historical facts, key results, and quotes, with appropriate hedging for open conjectures.
Commentary
- Collatz verification limit is approximately 2^71 ≈ 2.36 × 10^21, matching article; recent work may push higher but does not contradict [3].
- Article simplifies technical details (e.g., "almost all" for Tao) but uses appropriate popular science language without misleading.
- Euler's 1742 Goldbach quote paraphrased but faithful to historical context [6].
Sources used for verification
Academic/Peer-reviewed:
- The "bounded gaps between primes" Polymath project - arXiv
- The Millennium Prize Problems - claymath.org
- Yitang Zhang's Spectacular Mathematical Journey - ias.edu
Other reliable sources:
- Unheralded Mathematician Bridges the Prime Gap - quantamagazine.org
- Mathematician Proves Huge Result on ‘Dangerous’ Problem - quantamagazine.org
- Collatz conjecture - wikipedia.org
Fact-checked by Perplexity Sonar Pro on 2026-03-09
