LOG_10 x has an [early/large] cofr partial quotient of y. ##

[Don McDonald]

rem  >  user. parigp.LOG77 //log770.
date : 21.06.98  23:09

Seqfans,    26.05.03
method.
============
LOG(10) of x has an [early / large]  cf continued
 fraction partial quotient of y. ##

print continued fraction for logbase10(x), x = 2 to 500.
I couldn't find logbase 10
So I had to use ln(x) / ln(10).

if x is a multiple of 10 just print x instead
as an index marker.  log(10x) = 1. + log(x)
So it is covered before.
(I am looking for N*log(x) differs little from a whole number.)##
   i.e.  x^N = close to power of 10.

Look for large early partial quotients.
Write down the index x.

Go to myprog "SeriCalc4S".

Generate convergents to the chosen LOG(x) s.

THEN  5log63= 8.9967**,
2123log63 = 3819.999986,
74001*log63 = 133153 ##
 differs slightly from whole nos. ##

ALSO illustrating a real mathematical application of
Euclidean algorithm continued fraction convergents.

In BBC Acorn UK Basic64. (5-byte reals.)

don.mcdonald
-----------------ooooooooooooo

BEGIN, ENTER START NUMBER (Expression),  Q. END

?LOG63
Result--(1) 1.79934055  DISPLAY COMMA'S (,000)

          1.799,340,549,453,581,68

(2) 1.79934055  ƒCTD FRACTION,
 RATIO TO  DENOM 'A' (default 1)
 (ENTER NO. OR EXPRESSION) ?
 = 1
 CALCULATE B / A =1.79934055 / 1
 = 1.79934055  recipr = 0.55575916

CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction
continued fraction (very good -and convergents) for log_10 63.

1 1/1= 1
1 2/1= 2
3 7/4= 1.75
1 9/5= 1.80000000000000004
PREVIOUS LINE IS GOOD APPROXIMATION

59 538/299= 1.79933110367892968
1 547/304= 1.79934210526315797
PREVIOUS LINE IS GOOD APPROXIMATION

5 3273/1819= 1.79934029686641006
1 3820/2123= 1.79934055581723973
PREVIOUS LINE IS GOOD APPROXIMATION

34 133153/74001= 1.79934054945203448
PREVIOUS LINE IS GOOD APPROXIMATION

117 15582721/8660240= 1.79934054945359478
1 15715874/8734241= 1.79934054945358168
LIMIT , BIG ENOUGH


(3) 1.79934055
(4) 1.79934055  OPTION J. nx multiples
DISP. MULTIPLES AND SUBMULTIPLES OF X.

1*x= 1.79934055    x/n= 1.79934055
2*x= 3.5986811     x/n= 0.89967027
3*x= 5.39802165    x/n= 0.59978018
4*x= 7.1973622     x/n= 0.44983514
nx multiples 1.79934055/submult<ESC> 
5*x= 8.99670275    x/n= 0.35986811
6*x= 10.7960433    x/n= 0.29989009

log63/6 = .3 = 3/10.
63^10 = 9.85E17.?

Q. PROGRAM  SERIcalc4S  E N D. // don.mcdonald.


log 33 x
LOG 92  has partial.quotient.  380
LOG 152  has part.quotient.  324

LOG 168  part quot. 5936 (missed) **
LOG 231  has part.quotient.  338
LOG 261  has part.quotient.  265
LOG 269  has part.quotient.  318
LOG 276*    2335 ******
LOG 316      has part.quotient.  798
LOG 323      has part.quotient.  550
LOG 396*    3696 ****

LOG 398 *  missed.
LOG 468  has part.quotient.  850
LOG 494  has part.quotient.  1167
LOG

regards  /  don.
mcdonald

*gp             GP/PARI CALCULATOR Version 1.36
                 (Archimedes version)

\precision      = 28
\prompt         = ?

? print( cf( log(63) / log(10) ) )
%1 = [1, 1, 3, 1, 59*, 1, 5, 1, 34*, 117*, 1, 122*, 2, 10, 11, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1]
5log63,  2123log63,  74001*log63 differs slightly from whole nos. ##

? ?if
if(a,seq1,seq2)= if a is nonzero, seq1 is evaluated, otherwise seq2.

? for( x = 7,12, if( x%10, print( cf( log(x) / log(10) ) ),print(x) ))
                      ############^^^^^^^^^^^^########

[0, 1, 5, 2, 5, 6, 1, 4813*, 1, 1, 2, 2, 2, 1, 1, 1, 6, 5, 1, 83, 7, 2, 1, 1, 1]
[0, 1, 9, 3, 7, 2, 1, 19, 5, 1, 3, 6, 3, 2, 4, 5, 1, 1, 1, 1, 13, 1, 2, 1, 1, 15, 3, 8, 2]
[0, 1, 20, 1, 5, 1, 6, 2, 3, 1, 1, 8, 1, 2, 2, 5, 1, 1, 1, 1, 1, 3, 7, 4, 22, 2, 1, 1, 29, 1, 5]
10   ***  count marker
[1, 24, 6, 3, 2, 1, 1, 3, 1, 1, 1, 9, 1, 3, 1, 1, 1, 4, 1, 3, 1, 27, 1, 3, 5, 1, 2, 3, 42]
[1, 12, 1, 1, 1, 2, 3, 3, 2, 1, 7, 2, 1, 3, 2, 6, 5, 19, 2, 2, 1, 2, 1, 1, 4, 1, 6, 3, 3, 1]

? for( x = 2, 500, if( x%10, print( cf( log(x) / log(10) ) ),print(x) ))

[0, 3, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18, 1, 6, 1, 2, 1, 1, 4, 1, 42, 6, 1, 4, 2, 3, 1, 2]
[0, 2, 10, 2, 2, 1, 13, 1, 7, 18, 2, 2, 1, 2, 3, 4, 1, 1, 14, 2, 44, 1, 3, 1, 14, 2, 2, 1]
[0, 1, 1, 1, 1, 18, 1, 4, 2, 12, 1, 3, 1, 1, 2, 9, 2, 3, 2, 1, 3, 1, 1, 1, 85, 3, 2, 2, 4, 1, 1, 5]
[0, 1, 2, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18, 1, 6, 1, 2, 1, 1, 4, 1, 42, 6, 1, 4, 2, 3, 1, 2]
[0, 1, 3, 1, 1, 32, 1, 1, 278*, 1, 1, 2, 5, 1, 4, 1, 3, 7, 1, 1, 44, 1, 4, 2, 1, 1, 2, 8, 2, 1]             595*log6 = 462.999 994.
[0, 1, 5, 2, 5, 6, 1, 4813, 1, 1, 2, 2, 2, 1, 1, 1, 6, 5, 1, 83, 7, 2, 1, 1, 1]
...     much further on.

490
[2, 1, 2, 4, 4, 1, 1, 2, 7, 2, 1, 6, 1, 7, 4, 1, 2, 5, 4, 1, 1, 23, 1, 1, 1, 45, 1, 6, 1, 4, 2]
[2, 1, 2, 4, 17, 24, 2, 1, 112, 3, 1, 2, 1, 5, 2, 11, 1, 13, 1, 3, 7, 1, 2, 2, 1, 2, 1]
[2, 1, 2, 3, 1, 10, 4, 1, 8, 1, 3, 3, 1, 1, 2, 2, 1, 8, 2, 15, 1, 2, 2, 2, 4, 1, 1, 13, 1, 4, 1, 1]
[2, 1, 2, 3, 1, 3, 2, 1, 1, 1167*log494, 1, 3, 3, 5, 1, 4, 7, 1, 7, 1, 1, 1, 2, 2, 1, 57, 1, 5]
[2, 1, 2, 3, 1, 1, 1, 4, 6, 4, 1, 2, 1, 7, 9, 1, 2, 1, 1, 1, 1, 8, 1, 17, 6, 6, 6, 5, 2, 1, 1, 1]
[2, 1, 2, 3, 1, 1, 10, 1, 1, 10, 1, 4, 3, 5, 1, 1, 2, 1, 3, 3, 51, 1, 28, 1, 4, 1, 1, 4, 1, 1, 1]
[2, 1, 2, 3, 2, 2, 4, 144, 3, 1, 1, 1, 1, 3, 1, 565*, 1, 1, 11, 3, 1, 238, 1]
[2, 1, 2, 3, 3, 3, 4, 3, 1, 1, 3, 2, 5, 5, 11, 1, 11, 11, 1, 1, 1, 5, 3, 4, 4, 1, 2, 1, 4, 1]
[2, 1, 2, 3, 4, 1, 27, 3, 3, 1, 1, 193, 2, 1, 1, 1, 3, 5, 1, 100, 1, 1, 1, 2, 1, 3, 11, 1]
500
? \q.
*sp.