A new paper has dramatically advanced number theory by greatly refining what we know about the distribution of prime numbers.
The holy grail would be the twin prime conjecture which is that there are infinitely many pairs of prime numbers that are adjacent odd numbers. This hasn't been achieved, but the new paper does prove that there are infinitely many prime numbers within six hundred of each other, and that if something called the Elliot-Halberstam conjecture is true, that there are infinitely many prime numbers within twelve of each other.
As recently as May of this year it was considered a breakthrough when Yitang Zhang proved that there were infinitely many primes within 70 million of each other.
The Elliot-Halberstam conjecture is basically a weaker form of the generalized Riemann hypothesis, and both concern the frequency with which prime numbers are found in sequences of natural numbers.
Goldbach's Weak Conjecture
Meanwhile, Peruvian mathematician Harald Helfgott claims to have proved Goldbach's Weak Conjecture in 2013. Thus, he claims to have proved that: "Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum.)" and also as "Every odd number greater than 7 can be expressed as the sum of three odd primes." (The companion proof by example for numbers under about 10^30 is found here).
Goldbach's Strong Conjecture, that "Every even integer greater than two can be written as the sum of two primes," remains unproven. The weak conjecture is trivially implied by the strong conjecture, because one can always choose an even number which can be expressed as a sum of two primes and then add three, to express the corresponding odd number as a sum of three primes. But, since the three prime solutions to Goldbach's Weak Conjecture as proven, do not always include the number three, the converse does not flow from the Weak Conjecture. Empirically, the strong conjecture is true, at least, for every number less than 4*10^18.
While Goldbach's Strong Conjecture has not been proved, it was established in 1995 that every even number n greater than or equal to four can be expressed as a sum of at most six primes (which follows trivially from the just established proof of Goldbach's Weak Conjecture), and that "every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes) such as 100=23 +7*11.
Proof of Goldbach's Weak Conjecture makes it trivial to prove as a corollary that every even integer can be written as a sum of at most four primes, a significant improvement over the six prime result of 1995, but Harald Helfgott states in a letter quoted in the linked material that progress beyond four prime partitions of even numbers will be much harder. This is because every even number greater than 4 can be written as a sum of the prime number three (which is an odd prime) and an odd number, and every odd number can be written as a sum of at most three odd primes.
Previous coverage of this conjecture at this blog can be found here.
While this 2013 result has yet to be fully vetted, it will rival Fermat's Last Theorem (proposed in 1637 and first proven in 1995 in a documented and verified form) in notoriety if it is determined to be correct. Goldbach's conjecture was first stated in 1742.
The Riemann Hypothesis
The Riemann Hypothesis was stated in 1859 (and generalized later).
It has been known since 1997 that the generalized Riemann hypothesis implies Goldbach's weak conjecture. And, while the converse is not true, both the progress on the Twin Primes conjecture and the progress on Goldbach's Weak Conjecture, make a proof of the generalized Riemann hypothesis, which is the biggest prize in all of number theory because it implies so many other results, seem like something that could be achieved in my lifetime.
We are on the verge of experiencing a revolution in number theory that fills in the missing links necessary to prove myriad statements about the hidden underlying structure of our number system that can provide a unified foundation for further understanding in a field that has thus far been largely made up of isolated results that seem similar in character to each other but are not yet linked by any real unifying principal.