%I #36 May 05 2023 16:12:09
%S 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 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 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 Decimal representation of Bernays's number, 67^257^729.
%C This is a finite sequence of length:
%C 12669674935126608420432141630855015714031380133279087897111823021713
%C 56811328908882531121111469241905999472837913948238279755189743349761
%C 48523228801813277516107342082973093097725413191748277420852876334406
%C 94876293314725026209146791804598489379530361645466750631479593491258
%C 89808249942992766773762667299010546238077478887602330971928923721941
%C 72430386014378023796026916142427291438343856787929901324301858848058
%C 27529209171651712159083473169942927988975800558560613650527749524532
%C 75191774503837456493065661204570029626133921181521620538048404123145
%C 80067317493063106206968226133732232940295274157977766381479776103292
%C 42109055590354062378067741707188662030279086463891262509315257332626
%C 67660622430734107904225269523607105245934662799643886003767606189798
%C 57787550338482005946448422968364423839287672830452222083405995953816
%C 36203273931424615452013250308765219156613666060842449019621385654602
%C 20267721814801560898692089207050744121863093763466729360829090007340
%C 19845582687744823456294708029903891488031593815746468873765082722973
%C 55869028659436217274868023452405819990037705937486501551418694155825
%C 46884222339479672918917024200948456377272821591381189093132349850355
%C 90405444255979897423051268599606922301116055394691960425916429039897
%C 40352095868171539874185632233360706548132174778016724460684684331817
%C 19808367766356367096522727921316089451547342396490948067779940625178
%C 88020116160602011047647958441543061184800996681742861951458927608369
%C 31921303463436907590593465227992579980690076538677526802642563241223
%C 58778978568413308707865022089920596975426734290393003094530833538477
%C 51070253043054985292670186337562849238518822912544387932065661784941
%C 62666108221075583052234535354001732258294144569659354587932951541940
%C 4998441803274619168045467726087340720754974495397486708986
%C ... which is 1 greater than the abscissa of the common logarithm of the number.
%C From _Daniel Forgues_, Jan 14 2012: (Start)
%C The length 1.2669... * 10^1757 (shown above) itself has 1758 digits!
%C The last 100 digits of Bernays's number are 67^257^729 mod 10^100 =
%C 99736332723695669681470601458905407678415512345606116119671058321958
%C 37693587043524719438498607119427. (End)
%H 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 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 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 Efunda Engineering Fundamentals, <a href="http://www.efunda.com/math/exp_log/series_exp.cfm">Series Expansion of Exponential and Logarithmic Functions</a>
%H K. Podnieks, <a href="http://www.ltn.lv/~podnieks/finitism.htm">Towards a real Finitism?</a>
%F 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 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 (bc) /* bc script for cygwin bash shell or other UNIX environment */
%o /* Explicitly scale to 4000 decimal places */
%o i=10^4000
%o /* Calculate natural log of 100/67 */
%o b=100*i/67
%o c=i*(b-i)/(b+i)
%o x=c
%o s=0
%o for( j=1 ; x/j >0 ; j=j+2 ){
%o s = s + x/j
%o x = x*c/i*c/i
%o j /* progress mark */
%o }
%o s=2*s
%o /* Now s is the natural log of 100/67 */
%o /* Calculate natural log of sqrt(sqrt(10)) */
%o b=sqrt(10*i*i)
%o b=sqrt(b*i)
%o c=i*(b-i)/(b+i)
%o x=c
%o t=0
%o for( j=1 ; x/j >0 ; j=j+2 ){
%o t = t + x/j
%o x = x*c/i*c/i
%o j /* progress mark */
%o }
%o t=2*t
%o /* Now t is the integer part of 10^4000 * ln sqrt(sqrt(10)) */
%o ln10=4*t
%o /* ln10 is the integer part of 10^4000 * ln 10 */
%o ln67 = 2*ln10 - s
%o /* ln67 is the integer part of 10^4000 * ln 67 */
%o lg67 = ln67*i/ln10
%o /* lg67 is the integer part of 10^4000 * log_10 67 */
%o a=257^729
%o lgb = a*lg67
%o /* lgb is the integer part of 10^4000 * log_10 67^257^729 */
%o absc = lgb/i
%o /* absc is the abscissa of lgb, and its value is one less than the
%o number of decimal digits in Bernays's number */
%o mant = lgb - i*absc
%o /* Find number of digits in abscissa */
%o x=absc
%o for( nab=0 ; x>0 ; nab++ ) x = x/10
%o /* reduce the scale by nab */
%o mant = mant/10^nab
%o ln10 = ln10/10^nab
%o i = i/10^nab
%o /* find ln 10^mant */
%o lnmant = mant*ln10/i
%o /* calculate exp(lnmant) to get leading digits of Bernays's number */
%o fac=1
%o x=i
%o n=0
%o for( j=0 ; x/fac > 0 ; j++ ){
%o n = n + x/fac
%o x=x*lnmant/i
%o fac = fac*(j+1)
%o j /* progress mark */
%o }
%o /* display abscissa of log_10 ( Bernays's number ) */
%o absc
%o /* Display leading digits of Bernays's number.
%o ( Truncation is to avoid displaying round-off error )*/
%o n / 10^20
%K cons,nonn,fini
%O 1,1
%A Lewis Mammel (l_mammel(AT)att.net), May 02 2009