cp's OEIS Frontend

A160106 Decimal representation of Bernays's number, 67^257^729.

%I A160106 #36 May 05 2023 16:12:09
%S A160106 3,2,5,9,1,5,3,8,4,7,9,8,6,1,8,9,9,1,6,2,5,7,1,7,8,0,6,8,8,5,0,1,5,8,
%T A160106 8,1,1,2,0,8,6,2,1,1,5,0,5,8,7,7,8,0,7,7,6,4,5,0,7,2,8,7,0,2,5,0,4,8,
%U A160106 0,9,7,0,2,1,2,5,0,1,7,4,1,3,3,1,7,5,6,1,7,3,6,5,3,6,4,5,8,7,4,3,2,3,5,3,7
%N A160106 Decimal representation of Bernays's number, 67^257^729.
%C A160106 This is a finite sequence of length:
%C A160106 12669674935126608420432141630855015714031380133279087897111823021713
%C A160106 56811328908882531121111469241905999472837913948238279755189743349761
%C A160106 48523228801813277516107342082973093097725413191748277420852876334406
%C A160106 94876293314725026209146791804598489379530361645466750631479593491258
%C A160106 89808249942992766773762667299010546238077478887602330971928923721941
%C A160106 72430386014378023796026916142427291438343856787929901324301858848058
%C A160106 27529209171651712159083473169942927988975800558560613650527749524532
%C A160106 75191774503837456493065661204570029626133921181521620538048404123145
%C A160106 80067317493063106206968226133732232940295274157977766381479776103292
%C A160106 42109055590354062378067741707188662030279086463891262509315257332626
%C A160106 67660622430734107904225269523607105245934662799643886003767606189798
%C A160106 57787550338482005946448422968364423839287672830452222083405995953816
%C A160106 36203273931424615452013250308765219156613666060842449019621385654602
%C A160106 20267721814801560898692089207050744121863093763466729360829090007340
%C A160106 19845582687744823456294708029903891488031593815746468873765082722973
%C A160106 55869028659436217274868023452405819990037705937486501551418694155825
%C A160106 46884222339479672918917024200948456377272821591381189093132349850355
%C A160106 90405444255979897423051268599606922301116055394691960425916429039897
%C A160106 40352095868171539874185632233360706548132174778016724460684684331817
%C A160106 19808367766356367096522727921316089451547342396490948067779940625178
%C A160106 88020116160602011047647958441543061184800996681742861951458927608369
%C A160106 31921303463436907590593465227992579980690076538677526802642563241223
%C A160106 58778978568413308707865022089920596975426734290393003094530833538477
%C A160106 51070253043054985292670186337562849238518822912544387932065661784941
%C A160106 62666108221075583052234535354001732258294144569659354587932951541940
%C A160106 4998441803274619168045467726087340720754974495397486708986
%C A160106 ... which is 1 greater than the abscissa of the common logarithm of the number.
%C A160106 From _Daniel Forgues_, Jan 14 2012: (Start)
%C A160106 The length 1.2669... * 10^1757 (shown above) itself has 1758 digits!
%C A160106 The last 100 digits of Bernays's number are 67^257^729 mod 10^100 =
%C A160106 99736332723695669681470601458905407678415512345606116119671058321958
%C A160106 37693587043524719438498607119427. (End)
%H A160106 Robert G. Wilson v, <a href="/A160106/b160106.txt">Table of n, a(n) for n = 1..10000</a> (first 2223 digits from Lewis Mammel)
%H A160106 P. Bernays, <a href="http://doi.org/10.5169/seals-26602">Sur le platonisme dans les mathématiques</a>, L'Enseignement Mathématique 34 (1935), pp. 52-69. (See p. 61.)
%H A160106 P. Bernays, <a href="https://www.phil.cmu.edu/projects/bernays/Pdf/bernays13_2002-11-26.pdf">Platonism in mathematics</a>, translated from the French by C. D. Parsons.
%H A160106 Efunda Engineering Fundamentals, <a href="http://www.efunda.com/math/exp_log/series_exp.cfm">Series Expansion of Exponential and Logarithmic Functions</a>
%H A160106 K. Podnieks, <a href="http://www.ltn.lv/~podnieks/finitism.htm">Towards a real Finitism?</a>
%F A160106 Bernays's number is 67^257^729. The length and values of the sequence of its decimal representation is found by calculating its common logarithm by the formula, 257^729 * log_10(67) using an extended precision of 4000 digits. The number of digits of Bernays's number is given by the abscissa plus one, and the initial sequence is calculated from exponentiation of the mantissa.
%t A160106 nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[exp*Log10[base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[67, 257^729] (* and the last 100 digits computed by PowerMod[67, 257^729, 10^100] *) (* _Robert G. Wilson v_, Aug 09 2016 *)
%o A160106 (bc) /* bc script for cygwin bash shell or other UNIX environment */
%o A160106 /* Explicitly scale to 4000 decimal places */
%o A160106 i=10^4000
%o A160106 /* Calculate natural log of 100/67 */
%o A160106 b=100*i/67
%o A160106 c=i*(b-i)/(b+i)
%o A160106 x=c
%o A160106 s=0
%o A160106 for( j=1 ; x/j >0 ; j=j+2 ){
%o A160106 s = s + x/j
%o A160106 x = x*c/i*c/i
%o A160106 j /* progress mark */
%o A160106 }
%o A160106 s=2*s
%o A160106 /* Now s is the natural log of 100/67 */
%o A160106 /* Calculate natural log of sqrt(sqrt(10)) */
%o A160106 b=sqrt(10*i*i)
%o A160106 b=sqrt(b*i)
%o A160106 c=i*(b-i)/(b+i)
%o A160106 x=c
%o A160106 t=0
%o A160106 for( j=1 ; x/j >0 ; j=j+2 ){
%o A160106 t = t + x/j
%o A160106 x = x*c/i*c/i
%o A160106 j /* progress mark */
%o A160106 }
%o A160106 t=2*t
%o A160106 /* Now t is the integer part of 10^4000 * ln sqrt(sqrt(10)) */
%o A160106 ln10=4*t
%o A160106 /* ln10 is the integer part of 10^4000 * ln 10 */
%o A160106 ln67 = 2*ln10 - s
%o A160106 /* ln67 is the integer part of 10^4000 * ln 67 */
%o A160106 lg67 = ln67*i/ln10
%o A160106 /* lg67 is the integer part of 10^4000 * log_10 67 */
%o A160106 a=257^729
%o A160106 lgb = a*lg67
%o A160106 /* lgb is the integer part of 10^4000 * log_10 67^257^729 */
%o A160106 absc = lgb/i
%o A160106 /* absc is the abscissa of lgb, and its value is one less than the
%o A160106 number of decimal digits in Bernays's number */
%o A160106 mant = lgb - i*absc
%o A160106 /* Find number of digits in abscissa */
%o A160106 x=absc
%o A160106 for( nab=0 ; x>0 ; nab++ ) x = x/10
%o A160106 /* reduce the scale by nab */
%o A160106 mant = mant/10^nab
%o A160106 ln10 = ln10/10^nab
%o A160106 i = i/10^nab
%o A160106 /* find ln 10^mant */
%o A160106 lnmant = mant*ln10/i
%o A160106 /* calculate exp(lnmant) to get leading digits of Bernays's number */
%o A160106 fac=1
%o A160106 x=i
%o A160106 n=0
%o A160106 for( j=0 ; x/fac > 0 ; j++ ){
%o A160106 n = n + x/fac
%o A160106 x=x*lnmant/i
%o A160106 fac = fac*(j+1)
%o A160106 j /* progress mark */
%o A160106 }
%o A160106 /* display abscissa of log_10 ( Bernays's number ) */
%o A160106 absc
%o A160106 /* Display leading digits of Bernays's number.
%o A160106 ( Truncation is to avoid displaying round-off error )*/
%o A160106 n / 10^20
%K A160106 cons,nonn,fini
%O A160106 1,1
%A A160106 Lewis Mammel (l_mammel(AT)att.net), May 02 2009