Some very recent, current news. As everyone knows, there are multiple ways to add numbers to get a specific number. For example, you can get 5 by adding 2+3, 1+4, 1+1+3, and so on. While this may seem fairly trivial and useless, it actually comes up in many areas of math and physics, such as Abstract Algebra in the form of the partition function, p(n). Basically, the value of p(n) is the number of unique (up to isomorphism) ways you can add positive integers to form n, which is also a positive integer (if n <= 0, then p(n) is defined to be 0). For example, let us find p(5):

(1) 1+1+1+1+1=5

(2) 1+1+1+2=5

(3) 1+2+2=5

(4) 1+1+3=5

(5) 2+3=5

(6) 1+4=5

(7) 5=5

Thus we find that p(5) = 7. As you might be able to tell, as we go to higher values of n, this gets more and more complicated in terms of determining it this way, though you might hope that there is a simple pattern. These are the first 10 values for n=1,2,...10: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42. For n=1000, p(n)=24,061,467,864,032,662,473,692,149,727,991. This is quite a large number indeed, and there isn't really any immediate pattern found, though there is an explicit function for it of sorts, it is not so simply evaluated and uses an infinite summation. However, recently (ie. the news is about 3 days old), an algebraic formula has been discovered that involves no such infinite sums and is much more powerful in determining values of p(n), which is quite a discovery. The paper on it is authored by Jan Hendrik Bruinier and Ken Ono.

Here's a link to the results of the paper:

http://www.aimath.org/news/partition/brunier-ono.pdf