"Cremer number 2"

The last Cremer number produced a final Julia seed interval of width 7*2^-25, a bit too large for my liking. So I tried to calculate another one.

\[

a_0 = 2 \\

a_1 = 2 \\

a_n=2^{Q_{n-1}}

\]

also converges to a Cremer number (proof in the zipped file).

The maxima code to compute the convergents:

`cremer_init(a0,a1):=(`

listan:[a1,a0],

listpn:[a0*a1+1,a0],

listqn:[a1,1],

approx:[listpn[1]/listqn[1],listpn[2]/listqn[2]]

)$

cremer_iterate():=(

an:2^listqn[1], /* very fast growth */

listan:append([an],listan),

tmpqn:an*listqn[1]+listqn[2],

tmppn:an*listpn[1]+listpn[2],

listqn:append([tmpqn],listqn),

listpn:append([tmppn],listpn),

approx:append([tmppn/tmpqn],approx)

)$

denom2N(a,b,N) := (

/* converts two rationals a <= b comprising an interval into two rationals */

/* cdl <= cdr whose denominator is 2^N with cdl <= a <= b <= cdr */

if a < b then left:a else left:b,

if a > b then right:a else right:b,

left2:left * 2^N,

/* attn: floating point operations might be used internally */

cdl:floor(left2),

/* make sure the following verification works solely on rationals */

if (integerp(cdl) = false) then print("Error 1"),

/* verify that it is valid */

/* the check then works again on rationals */

if ( (cd1 / (2^N)) > left) then print("Error 2"),

if (cdl <= (2^63 - 1)) then print(" cdl usable as int64_t") else print(" ATTN: cdl larger than int64_t"),

if (cdl <= (2^52 - 1)) then print(" cdl usable as double") else print(" ATTN: cdl might need more precision than double"),

right2:right * 2^N,

if mod(right2,2^N) = 0 then cdr:floor(right2) else cdr:1+floor(right2),

if (integerp(cdr) = false) then print("Error 3"),

/* and check if a valid enclosement */

if ( (cdr / (2^N)) < right) then print("Error 4"),

if (cdr <= (2^63 - 1)) then print(" cdr usable as int64_t") else print(" ATTN: cdr larger than int64_t"),

if (cdr <= (2^52 - 1)) then print(" cdr usable as double") else print(" ATTN: cdr might need more precision than double"),

display(cdl),

display(cdr)

)$

using

*cremer_init(2,2);*

*cremer_iterate();*

*cremer_iterate();*

*cremer_iterate();*

*denom2N(approx[1],approx[2],48);*

With that definition I could compute P5/Q5 (values in the zipped file), the difference to P4/Q4 being smaller than 10^{-1300}. I guess that's why Brjuno numbers are said to be badly approximated by rationals - this one is the opposite, P4/Q4 = 11269/4610 is already pretty close.

Using those rationals (outward converted to a denominator of 2^48) to compute a final c interval (after conjugation) for z^2+c resulted in:

\[

c=( [-186176550019915~..-186176550019910]~+~i*[93330855667149~..93330855667158] )~*~2^{-48}

\]

which has only a width of 9 * 2^-48.

Next steps now: Image generation, proof extension to obtain further values and trying to formalize the proof to be verifiable by an automated proof checker system.