Some physics thoughts ... a) You can reduce the energy required to move things into space by exchanging with moon rocks. Using a few Moravec- style rotating tethers, a bucket brigade can be established that imports moon rocks, while establishing a CO2 atmosphere on the moon. b) Dan's nit about the gravity field being non-parallel. This was dismissed because of the exponential falloff of gas density with height, "so the extra weight of the stuff at the top is negligible". But the effect of that extra weight is multiplicative, not additive: The extra weight needs extra supporting gas in the layer below, etc., so the overall result could be larger. c) The halving-density height of air is about 3.5 miles. A 35 mile deep well would have air that's a little denser than water at the bottom. (Might be interesting to jump into!) The density near the center of the earth would exceed black hole numbers, except that the gas laws wrt compression and pressure fail. d) If you carve out a small hole for yourself in the center of the earth, it's true that the gravitational force on you will balance out to near 0 (ignoring a non-spherical earth). But the pressure might still be important. (Or we'd expect bubbles to form in the middle of fluid planets like Saturn.) e) I think the halving-height of a gas depends on the molecular weight, and that pure CO2 would have a smaller halving height. ----- and some GF2 thoughts ... f) The sum-of-divisors formula should work in GF2. g) I read MLB's problem as "find sets of GF2 prime polynomials that xor to 0". No biggie: 11111 x 11001 x 1101 x 1011. More generally, there are about 2^D / D prime polys of degree D, and 2^that possible sets. Many of those sets must xor to the same value (only 2^(D+1) possible xor results), and thence many 0s. ----- re 10^3 + 1 = 7.11.13: If you drop the "consecutive" requirement on the primes, I think you can come up with lots more examples: 12^3 + 1 = 1729 = 7.13.19 Rich