One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Twin Prime Conjecture — Are There Infinitely Many Close-Packed Prime Pairs?
Are there infinitely many pairs (p, p+2) where both are prime? The case is included in de Polignac’s 1846 conjecture. In 2013 Yitang Zhang proved the first finite upper bound for infinitely recurring consecutive-prime gaps, without reaching gap two.
Test small cases immediately with a drawing, divisors, or colored pencils.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesNo formal prize. The post-2013 bounded-gap theorems must be distinguished from the original gap-two conjecture.
See the puzzle visually
After Zhang announced a gap of 70 million in 2013, the unconditional bound fell to 246 within months. Reaching the final gap of 2 would prove the Twin Prime Conjecture.
Problem statement
Twin primes are pairs (p, p+2) where both are prime. Is the set {p prime : p+2 prime} infinite?
The story of this puzzle
Zhang had worked outside academia after his doctorate, but he had taught at the University of New Hampshire since 1999 when he submitted his paper in 2013. He proved that consecutive-prime gaps below 70 million occur infinitely often. Polymath and independent methods of Maynard and Tao sharply improved and extended bounded-gap results. The unconditional bound 246 is still logically far from the target gap two; numerical improvement alone does not force the twin-prime conjecture.
Try it yourself
Find the first 100 twin prime pairs starting with (3,5). Up to 100,000 there are 1,224 pairs. Code a search to 1M and plot the distribution.
Beyond MathVoyage
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