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