Lola Thompson: Mind the Gaps Between Primes

The Mystery of Prime Numbers

• 💡 Prime numbers are considered the building blocks of integers, similar to how DNA is the building block of life.
• 🔍 Despite appearing randomly distributed, primes exhibit intriguing patterns, such as their alignment on specific diagonals in a spiral.

Cautionary Tales in Prime Pattern Seeking

• ⚠️ Fermat's conjecture (2^(2^n)+1 are all prime) failed for n=5, demonstrating that patterns observed in small numbers may not hold universally.
• 🤡 So-called "discoveries" by individuals like Solog, claiming primes fall into specific columns (mod 6), are trivial observations explained by basic divisibility rules.
• 🧐 It is crucial to maintain skepticism when searching for patterns in primes, as they can be either untrue or trivially explained.

The Quest for Bounded Gaps Between Primes

• ♾️ There are infinitely many primes, a fact proven by Euclid, and the sum of their reciprocals is infinite.
• 📉 The proportion of primes among integers decreases as numbers get larger, approaching 0%, despite their infinite quantity.
• 👯 The Twin Primes Conjecture posits there are infinitely many pairs of primes that differ by exactly two.

Breakthroughs by Zhang, Maynard, and Tao

• 🚀 In 2013, Yitang Zhang stunned mathematicians by proving there are infinitely many pairs of primes differing by at most 70,000,000.
• 🛠️ Zhang's work utilized sieve methods, including a modified GPY sieve, to identify numbers likely to be prime and close together.
• 🤝 Following Zhang, James Maynard and Terry Tao developed a method for admissible k-tuples, showing that at least 'm' numbers in a k-tuple can be prime.
• 🎯 The Polymath8 project, a crowdsourced effort, further reduced the maximum gap between infinitely many prime pairs to 246, using Maynard and Tao's ideas.

Applications and Further Research

• 🧩 The methods for bounded prime gaps have been applied to irreducible polynomials, proving analogous results for gaps between them.
• 🔢 Research by Thompson and co-authors resolved Sirpinsky's question (1961) about digit sums of consecutive primes.
• ✅ They demonstrated that for any base, there are arbitrarily long runs of consecutive primes where the digit sum is constant, increasing, or decreasing, without relying on unproven conjectures.