==> pickover/pickover.16.s <== ------------------------- In article <1992Oct30.175102.142177@watson.ibm.com> you write: : 1. Are there any undulating square numbers? 11^2 = 121 : 2. Are there any undulating cube numbers? 7^3 = 343 (yes, I know they're short, but they qualify!) -- Michael Neylon aka Masem the Great and Almighty Thermodynamics GOD! // | Senior, Chemical Engineering, Univ. of Toledo \\ // Only the | Summer Intern, NASA Lewis Research Center \ \X/ AMIGA! | mneylon@jupiter.cse.utoledo.edu / --------+ How do YOU spell 'potato'? How 'bout 'lousy'? +---------- "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L ------------------------- In article <1992Oct30.204134.97881@watson.ibm.com> you write: >Hi, I was interested in non-trivial cases. Those with greater >than 3 digits. Award goes to the person who finds the largest >undulating square or cube number. Thanks, Cliff 343 and 676 aren't trivial (unlike 121 and 484 it doesn't come from obvious algebraic identities). The chance that a "random" number around x should be a perfect square is about 1/sqrt(x); more generally, x^(-1+1/d) for a perfect d-th power. Since there are for each k only 90 k-digit undulants you expect to find only finitely many of these that are perfect powers, and none that are very large. But provably listing all cases is probably only barely, if at all, possible by present-day methods for treating exponential Diophantine equations, unless (as was shown in a rec.puzzles posting re your puzzles on arith. prog. of squares with common difference 10^k) there is some ad-hoc trick available. At any rate the largest undulating power is probably 69696=264^2, though 211^3=9393931 comes remarkably close. --Noam D. Elkies ------------------------- In article <1992Oct30.175102.142177@watson.ibm.com>, you write... >1. Are there any undulating square numbers? > Other than the obvious 11**2, 22**2, and 26**2, there is 264**2 which equals 69696. >2. Are there any undulating cube numbers? > Just 7**3 as far as I can tell, though I'm limited to IEEE computational reals. PauL M SchwartZ (-Z-) | Follow men's eyes as they look to the skies v206gb6c@ubvms.BitNet | the shifting shafts of shining pms@geog.buffalo.edu | weave the fabric of their dreams pms@acsu.buffalo.edu | - RUSH -