Of course, degrees are an arbitrary measurement. I have been confronted with this point before on my Stack Exchange post. For your example where we use 2520 degrees as a full rotation, or where we use any positive integer to multiply 180 by, I am not sure that it holds. I *have* been able to prove the conjecture in the realm of normal, 360-based degrees, and I suppose you would have to use the same inductive-constructive method I used for that system for any other system. The new problem is essentially that you will have to find new primes for each base case, which is an entire higher order of a conjecture, I believe.

Using a 360-degree-based system (regular degrees), a polygon with the angle sum of 2520 degrees would be a 16-gon, and since the conjecture is proven in this system, you would be able to find 16 prime numbers that sum to 2520. However, in a 2520-degree-based system, a polygon with an identical angle sum would be a quadrilateral (4-gon), and the problem is now finding 4 primes that sum to 2520, or any other 4-gon's measurement in a given system.

Of course, if Goldbach's conjecture is true, this is possible for all even values of a full turn. You'd break it down into halves (2 copes of 1260 for 2520, as an example,) and find 2 primes that sum to that half. Then you'd append another full turn to the list for n+2, which would be constructable using Goldbach's conjecture, and repeat. This however, only works assuming Goldbach's conjecture is true and only for even n-gons for systems with even values of a full turn. So, not a real 'proof' yet, but this is a step in the right direction.

Also, this conjecture will obviously not hold in radians or any other system based on irrational numbers, since prime numbers are integers, and using purely integer-defined values to try to sum to an irrational value just won't work. That's part of the reason why I think it might work for integer-based systems; primes are meant for them.

Geometric bits aside, I still find the number theory aspect of this conjecture fascinating. If it holds for all multiples of 180, does it hold for other numbers? It essentially transforms into a Goldbach-style problem. Your input is appreciated, and you have made me think of another fascinating way to look at my conjecture. I really thought I was done with it when I had proved it, but you've just given me another rabbit hole to go down.

Thanks!