## JSH: Puzzling through number theory?

With the success of the approach of presenting a major number theory issue as a puzzle I've began to consider that the problem with my prior approach has been that I've not given people the opportunity for intellectual challenge. Simply explaining mathematics may not be nearly as enticing as puzzling through.

So might that help in other areas? Conceivably here's another area for those who wish to match wits with me.

Find a prime counting function P that counts functions by first counting composites and subtracting that count and 1 from the total count to get primes, which only counts composites not already counted.

That is, Legendre's method rather inefficiently counts ALL composites at each prime that have that prime as a factor, which is dumb!!! Why, for instance, if you're counting composites divisible by 3, should you also count evens? It's WACKY!

Can you find a more efficient prime counting method than Legendre's which more smartly counts ONLY those composites at each prime that have not ALREADY been counted?

As a check, how hard is that as a puzzle? Should a smart, say, undergrad be able to succeed? How about a grad student?

If you think the problem is easy, great! Give the function in reply. If you have no clue, here's a clue.

Hint: The requirements force a P(x,y) function, so you end up with a multi-dimensional prime counting function instead of a pi(x), single variable one.

Can anyone solve the puzzle on their own? If you get completely lost, of course, the answer is on my math blog. Given lots of places as I've explained, and explained and explained.

But if I can figure it out, can you?