# New Breakthroughs in Prime Number Theory

# The Twin Prime Conjecture

First up, on April 17th (my birthday, no less), the journal *Annals of Mathematics* received a submission from Yitang Zhang of the University of New Hampshire purporting to at least partially prove the twin prime conjecture. The paper (available for preview here if you have access) will appear in an upcoming issue, but it is already creating a stir in the mathematics community. Building upon earlier work by Goldston, Pintz, and Yildrim (GPY), Zhang’s paper demonstrates that for any integer N less than 70 million, there are infinitely many pairs of primes that differ by N.

More at these links:

- Yitang Zhang Proves ‘Landmark’ Theorem in Distribution of Prime Numbers | Simons Foundation
- Bounded Gaps Between Primes | The n-Category Café
- Twin Prime Conjecture — from Wolfram MathWorld
- K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım
- Goldbach Variations | Roots of Unity, Scientific American Blog Network

# The Ternary Goldbach Conjecture

But wait, there’s more. H. A. Helfgott claims to a proven the ternary Goldbach conjecture.

In its most basic form, Goldbach’s conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. The ternary Goldbach conjecture, also known as the weak or odd Goldbach conjecture, states that every odd number greater than 5 can be expressed as the sum of three primes (with repeats allowed). For example, 7 can be written as 2+2+3.

More at these links:

- Goldbach Conjecture — from Wolfram MathWorld
- The Prime Glossary: Goldbach’s conjecture
- Goldbach Variations | Roots of Unity, Scientific American Blog Network
- Cracking Goldbach’s Conjecture
- On equivalent forms of the weak Goldbach conjecture | The Aperiodical