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:

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:


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