holy cow, that article is like 95% filler. this is the relevant part:

It’s only now, however, that Ono and his teammates have noticed something incredible: that “the prime numbers […] are the solutions of infinitely many special ‘Diophantine equations’ in well-studied partition functions,” they explain in their new paper. 

for example, the integer partitions of 4 will be 3+1, 2+2, 2+1+1, and 1+1+1+1

Hmmm...ok, sure...

That’s because partitions have a natural connection to a type of equation known as Diophantine equations – equations like Pythagoras’s or Markov’s, for which there are multiple if not infinitely many sets of rational solutions.

Diophantine equations are special because the interesting results are integer. For Pythagoras that would be something like 3, 4, 5 (3^2^ + 4^2^ = 5^2^).

The story seems to be implying some connection between adding integers and prime numbers, but it never gets anywhere close to explaining that connection.