Well, can Propp's sequences f[n] = SquareFreePart( f[n-1]+f[n-2] ) ever (with suitable starting integers) diverge to infinity without instead falling into a finite length cycle? Here SquareFreePart(X) means the product of all the primes that had odd powers inside X's factorization, e.g. SquareFreePart( 2*3*3*5*7*7*7*7 ) = 2*5. If we model the X as "random" then with chance 1/4 they contain a factor of 4 that will be removed, with chance 1/9 they contain a factor 9, etc. So ln(X) is going to decrease due to these removals, by an expected additive amount DELTA with DELTA <= SUM_{k>1} k^(-2) * ln(k^2) = 1.875096508631687507405148... The expected decrease amount also is lower bounded by the same sum but summed over prime k only: DELTA > 0.9861. Also note, the "probability distribution" of DELTA has finite variance. Meanwhile, ln(X) is going to grow due to the Fibonacci-like recurrence, by some amount presumably less than ln(2) = 0.693147180559945 in "expectation." A lower bound on the expected decrease amount is clearly greater than an upper bound on the expected increase amount. Therefore by the strong law of large numbers" of probability theory, it "follows" that increase of ln(f[n]) to infinity is impossible. This of course would only be a proof if the f[n] really behaved "randomly," which is bogus, but it is enough to make the "it always cycles" conjecture plausible. I tend to doubt Propp's stronger conjecture that only one particular limit cycle is possible; and to disprove that, it would suffice to find a single counterexample starting 2-tuple. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)