Are there infinitely many prime pairs like 11 and 13 or 17 and 19 twins and whats the latest research
Twin primes: , , , , ...
The Twin Prime Conjecture says there are infinitely many pairs of primes with difference 2.
Recently (2013), Yitang Zhang proved theres a bound such that infinitely many prime pairs differ by at most . Then the Polymath project reduced this to 246. And Maynard showed theres infinitely many prime pairs with gap without assuming the Elliott-Halberstam conjecture.
But theres still a gap between 246 and 2. Is there any hope of reducing it to 2?
Also: whats the current status of the conjecture? Is it "likely true" based on heuristics? I heard the Hardy-Littlewood conjecture gives a density estimate for twin primes.
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1 Answer
I would solve this by writing the assumptions and the target on separate lines first. That usually reveals which theorem is actually needed.
A good structure is:
- state the definitions involved;
- transform the expression without skipping algebra;
- check edge cases such as zero, negative values, or boundary points;
- substitute the result back into the original question.
This may feel slower, but it prevents the most common math-answer problem: getting a plausible expression that does not actually satisfy the original conditions.
Could you add one more line on the condition where this method fails?
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