A conjectured good rational approximations to the cubic irrationality of the real root of, 2 2 3 64 + 144 x (2 c + 1) + 108 x (3 c + 3 c + 1) + 27 x (2 c + 1) = 0, for positive intgers c, with amazing irrationality measures as c gets bigger By Shalosh B. Ekhad Consider the real root of the cubic equation (with a positive) 2 2 3 64 + 144 x (2 c + 1) + 108 x (3 c + 3 c + 1) + 27 x (2 c + 1) = 0 that according to Cardano equals 1/3 5 4 3 2 1/2 2 2 1/2 2 4 (c (-54 c - 162 c - 180 c + 2 c ((c + 1) ) - 90 c + ((c + 1) ) 1/3 3 2 2/3 / - 19 c - 1)) /(3 (2 c + 1)) + 2 c (9 c + 18 c + 11 c + 2) 4 / (3 / 5 4 3 2 1/2 2 (2 c + 1) (c (-54 c - 162 c - 180 c + 2 c ((c + 1) ) - 90 c 2 2 1/2 1/3 4 (3 c + 3 c + 1) + ((c + 1) ) - 19 c - 1)) ) - ------------------ 3 (2 c + 1) Consider the recurrence satisfied by the contour-integral, around x=0, of th\ e function /(c x + 1) (1 + (c + 1) x)\n |-------------------------| \ x / ---------------------------- 2/3 x let's call it X(n) that in Maple notation is ((c*x+1)*(1+(c+1)*x)/x)^n/x^(2/3) According to the famous Almkvist-Zeilberger algorithm, it satisfies the recu\ rrence X(n) = 9*n*(2*n-1)*(2*c+1)/(3*n-1)/(3*n+1)*X(n-1)-9*n*(n-1)/(3*n-1)/(3*n+1)*X(n -2) Let a(n), b(n), be the sequence that satisfy that recurrence with the initial conditions a(0) = 0, a(1) = 1 b(0) = 1, b(1) = 0 We conjecture that the limit of a(n)/b(n) as n goes to infinity is the above\ -mentioned cubic irrationality, namely the real root of 2 2 3 64 + 144 x (2 c + 1) + 108 x (3 c + 3 c + 1) + 27 x (2 c + 1) = 0 Not only that, these seem to give effective irrationality measures, that sta\ rting with c=4 are better than Liouville's irratioanlity measure of 3 The estimated irrationality measure for the case c=4 is 2.91671189692388132500469650131481711519220015086338875796898785652487333243\ 7465334476698221121493387 Let's make c bigger and see how the effective irrationaliy measure shrinks For c=, 11, the irrationality measure is estimated to be 2.62142830520850111722825132771928190392903298875791069032549231209765301155\ 6238400702347077250011859 For c=, 101, the irrationality measure is estimated to be 2.35373664843180663073757986375881023985617960073296259684672903697139392716\ 1655887277018248693710482 For c=, 1001, the irrationality measure is estimated to be 2.24435656427019258831196836391360623093579999326384378403454426981382307673\ 2081522524069181245322821 For c=, 10001, the irrationality measure is estimated to be 2.18624329935829019141483121953750310542157530895252891118020698341682766580\ 0235444966032459245702170 For c=, 100001, the irrationality measure is estimated to be 2.15044659611568879165606626307703764750110409260728203791726569488432883345\ 1297254010213171141808498 as you can see, it seems to go down to 2