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From a 70-Million Bound to Just 246: The Electrifying Story of Prime Gaps

For centuries, the twin prime conjecture seemed untouchable. Mathematicians couldn't even prove that any finite gap between primes occurs infinitely often.
Then, in 2013, a stunning breakthrough by Yitang Zhang proved the gap was less than 70,000,000. In less than a year, a global collaborative effort slashed that number to 246.

How did this happen? What new ideas unlocked a problem that was stuck for centuries? Dive into the full story, from the GPY sieve to the Maynard-Tao breakthrough, in the attached PDF.

#Number_Theory #Unsolved

The Atom of Addition: Are We "One Factor" Away from Solving Goldbach's Conjecture?
Is every even number greater than 2 the sum of two primes? For nearly 300 years, this simple question has defied a mathematical proof, despite computers verifying it up to $4 \times 10^{18}$. But we are tantalizingly close!
In 2013, Harald Helfgott proved the "weak" version for three primes, and Chen's Theorem already shows that we are just one prime factor away from the ultimate goal.
Why is that final step so incredibly hard to take? Dive into the full story of this beautiful mathematical mystery in the attached PDF!

#Number_Theory #Unsolved