There's another well-known concept of "positive" for algebraic numbers. I think it's named "totally positive", and it means the number is positive (in the reals) for all choices of embedding: I.e., all conjugates are positive. 1+sqrt2 fails here because the alternate embedding sqrt2 -> -1.414 gives a negative value (-.414). But 2+sqrt2 is totally positive. As is 2-sqrt2. It seems that totally positives are closed under + and *. I don't know what you do if there are complex embeddings. (cbrt2) [Puzzle: Identify interesting sets of complex numbers that are closed under + and *. (Right-half-plane works for +, fails for *.) Find a natural condition to exclude the trivial answer "start with any transcendental and close under +*".] ---- Two recent astronomy goodies: http://www.nso.edu/press/AO/ao.html Adaptive Optics View of the Sun Astronomers have developed the flex-mirror idea for a solar telescope. The hard problem is lack of sharp image contrasts to drive the servo-algorithm. The site has a couple of short movie clips of the surfce of the sun boiling, with convection cells a few hundred miles across. Must be some kind of magnetic viscosity. With a surfce gravity of 38G, you'd think the surface would be nearly flat. [Puzzle: Design a solar probe that will last for, say, a day. And figure out a way for it to communicate its measurements.] http://www.sciencedaily.com/releases/2003/06/030619075759.htm Speed of Gravity A few months ago, a group announced they'd measured the speed of gravity by looking at starlight bent by Jupiter's gravity. The above note is another group saying 'Tain't So. --- Hilarie and I have been enjoying the (solitaire) game Chromatron. "The first batch of puzzles is free." --- One of my favorite word puzzles is finding "consecutive" words, where the last letter is incremented to make a new word. CONVEX CONVEY TARN TARO TARP COMER COMES COMET MODEL MODEM CLOSER CLOSES CLOSET POSER POSES POSET (if we allow a bit of math jargon) The problem is to find more examples. The challenges are longer words, and longer sequences. I'd also like a true wraparound example, with Z incremented to A and the carry propagated to the next letter: OZ PA --- The GIMPS crowd (Mersenne prime search) recently had a false alarm. Someone's machine reported a final Lucas residue of 0. Woltman & co. insisted on keeping quiet until the double checks were completed, and the policy proved out. (I would shout it from the treetops. Oops.) Two different checking machines had nonzero final residues that matched each other. GIMPS errors happen: the GIMPS players run their machines hot (for speed), and O(1%) of the residues turn out wrong when double checked. And there's also the floating point used in the arithmetic, which occasionally has an undetected "overflow". [Floating point values are mapped to the nearest integer as part of the FFT multiplication.] But it's hard to imagine an arithmetic bug that would produce a 0 final residue (probability verging on infinitesimal). George has found another possible explanation, based on a NaN sneaking into the FFT somehow, and propagating. He's adding more checking of intermediate results to the program. --- New ECM (Elliptic Curve Method) factoring record: M997 (2^997-1) has the 57-digit divisor 167560816514084819488737767976263150405095191554732902607 found by Bruce Dodson. The cofactor is a probable prime. Rich rcs@cs.arizona.edu