Fun Math Problem.

I just finished a really fun elementary math problem that I needed for my work; thought I’d share the answer. The problem is this: suppose you have 8 positive numbers (p,q,r,s,t,u,v,w), all co-prime besides possibly being equal. How many different types of equality ensure:

p+q+r+s-t-u-v-w = 0          (1)

For example one solution is (p=t, q=u, r=v, s=w). The answer is related to a rotating-wave” or “secular” approximation for a many-particle density matrix in quantum dynamics. We can use sets of numbers which obey this condition to dramatically accelerate our many-body quantum dynamics calculations and treat correlation energy.

It turns out that the following expression of Kronecker deltas is 1 for every p,q,r,s… that satisfies (1) and zero otherwise, as you can easily verify. I’ll leave the derivation as an exercise to the reader :).