A collatz-inequality related to the waring-problem; a further question
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From: Gottfried Helms on 2 Jul 2010 06:20 In my own discussions of the cycle-problem, but also in the papers of R.Steiner, J.Simons & B.de Weger in their treatments of the "circuit"- or "1-cycle"-problem a certain equality in integers is required to make a 1-cycle possible, but heuristically that equality does not hold. This is 3^N - 1 ------- = 2^(S-N) (*1) 2^N - 1 where N and S are required to be positive integers>1, (for more background see below) Call the fraction on the lhs "cyc(N)" for the following. === a bit background =============================== This formula is based on a compressed notion of the collatz- transformation a1 = (3*a0 + 1)/ 2^A0 on odd(!) natural numbers a0,a1 and A0 the positive integer exponent, which allows oddness of a1. If more than one of such transformations are concatenated, then N is the number of such steps and let L=N-1 and S is the sum of the A0,A1,...A_L exponents. If a sequence of ascendent steps (A_k=1) are followed by exactly one descendent step (A_L >1) then this forms a trajectory having many steps upwards, then one peak and then one step downwards. Such a trajectory can be called a "1-peak-trajectory", and if a_N = a_0 then the trajectory forms a cycle and having one peak we may call it "1-cycle". It was proven by R.Steiner, that there is no such integer S for N>1 and thus no 1-cycle can exist. (J.Simons & B.de Weger improved that argumentation and extended the nonexistence-proof based on this up to 68-cycle, which is the concatenation of up to 68 1-peak- trajectories, each having arbitrary length, finally forming a cycle). ====== [end background] ========================== But while R. Steiner shows, that there is no integer S to build a proper perfect power of 2 equal to cyc(N) in (*1) we find empirically that there is a much sharper condition which can be tested to be satisfied or not. Even more, this condition is then related to another unsolved conjecture in numbertheory, namely the (unsolved) detail in the Waring-problem, to prove that 3^N 3^N frac( ----- ) < 1 - ----- (*2, conjectured) 2^N 4^N If this condition can be proven to be always true (for N greater than some small constant) then this implies the impossibility of the above equation (*1) (in integers), and thus the impossibility of the 1-cycle. ----------------------------------------- Inequality (*2) can be reformulated into 3^N 3^N 3^N ----- + ----- < ceil(---) (*2a) 2^N 4^N 2^N While (*2a) gives an upper bound, assuming symmetry as also indicated in [mathworld] we have also a lower bound: 3^N 3^N 3^N floor( --- ) < --- - --- (*2b) 2^N 2^N 4^N Thus we have the bracketing inequality (*2c, lhs and rhs inequalities conjectured by the waring-problem) 3^N 3^N 3^N 3^N 3^N 3^N 3^N floor(-----) < --- - --- < ---- < --- + --- < ceil( --- ) 2^N 2^N 4^N 2^N 2^N 4^N 2^N where the most-left ond the most right inequalities are subject of the Waring-problem-conjecture. Here it is important that the brackets of floor() and ceil() form an interval of 1, thus all values in between are trivially non-integer. The lhs of (*1) fits *into* this brackets; by some simple rewriting of cyc(N) we find that 3^N 3^N - 2^N cyc(N) = --- + --------- (*3.1) 2^N 4^N - 2^N and 3^N 3^N 3^N --- < cyc(N) < --- + --- (*3.2) 2^N 2^N 4^N But if the lower and upper bounds in (*2c) hold, cyc(N) cannot be integer (this is stronger than the lemma of R.Steiner, that cyc(N) cannot be a perfect power of 2). This way the solving of the conjecture (*2) in the Waring-problem automatically solves the 1-cycle-question in the Collatz-problem. --------------- Well, I've mentioned this fact already earlier, for instance in the my discussion in [cycles] and recently found it for instance discussed by S.Finch in [fi03]. But still I don't see this direct connection in discussions and common online-resources. But the extremely simple formulation of the conjecture in the Waring-problem looks as if it could be attacked more elementary than the 1-cycle-aspect in the Collatz-problem for which R.Steiner employed heavy-duty weapons from transcendental numbertheory, such that J Lagarias considered in his 1996-article: "The most remarkable thing about Theorem J <the Steiner-proof,G.H.> is the weakness of its conclusion compared to the strength of the methods used in its proof" Steven Finch in chap 2.30.1 "powers of 3/2 modulo 1" in [fi03] mentions the relation of that 1-cycle-problem with the Waring-problem, citing the inequality (*2.c) in the fractional-parts-notation and mentions a couple of authors which dealt with that question (among them also K.Mahler) without getting a final proof for it. (We find this all mentioned in Lagarias too, but not with that explicite connection as in (*2c) and (*3b)) So also this seemingly simple one will in fact be a difficult one. For some reason I looked at the taylor-series for the logarithms of the terms in (*2c) (when taken as functions of N) and cyc(N) If I denote that terms (including the cyc(N)-function) in some memorizable function-notation: fl_32(N) < w_low(N) < pow32(N) < cyc(N) < w_up(N) < ce_32(N) (*4) then the ln(cyc(N)) and ln(w_up(N)) come out to have taylor-series involving the bernoulli-numbers (which alone is surprising and triggering) To relate this to the integer(!) bounds by floor and ceil it would be interesting, if we could say whether there is some special with logs of consecutive *integers* or of "floor"s / "ceil"s. But here I don't have any idea. Is there something useful known about this which could be related to this problem? Gottfried Helms --------------------------------------------- [cycles] http://go.helms-net.de/math/collatz/Collatz061102.pdf an extended update is in progress... [lag96] "The 3x + 1 Problem and its Generalizations" J Lagarias, pg 28 online at: http://www.cecm.sfu.ca/organics/papers/lagarias/ [fi03] "Mathematical Constants", Steven Finch, Chap 2.30.1: "powers of 3/2 modulo 1" (pg 194) (interesting preview possible via google-books: http://books.google.de/books?id=Pl5I2ZSI6uAC&lpg=PP1&pg=PA194#v=onepage&q&f=true ) [mathworld] "Power Fractional Parts" Eric Weissstein online at: http://mathworld.wolfram.com/PowerFractionalParts.html |