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

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, 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, 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

Offset

1,1

Comments

This is a finite sequence of length:

12669674935126608420432141630855015714031380133279087897111823021713

56811328908882531121111469241905999472837913948238279755189743349761

48523228801813277516107342082973093097725413191748277420852876334406

94876293314725026209146791804598489379530361645466750631479593491258

89808249942992766773762667299010546238077478887602330971928923721941

72430386014378023796026916142427291438343856787929901324301858848058

27529209171651712159083473169942927988975800558560613650527749524532

75191774503837456493065661204570029626133921181521620538048404123145

80067317493063106206968226133732232940295274157977766381479776103292

42109055590354062378067741707188662030279086463891262509315257332626

67660622430734107904225269523607105245934662799643886003767606189798

57787550338482005946448422968364423839287672830452222083405995953816

36203273931424615452013250308765219156613666060842449019621385654602

20267721814801560898692089207050744121863093763466729360829090007340

19845582687744823456294708029903891488031593815746468873765082722973

55869028659436217274868023452405819990037705937486501551418694155825

46884222339479672918917024200948456377272821591381189093132349850355

90405444255979897423051268599606922301116055394691960425916429039897

40352095868171539874185632233360706548132174778016724460684684331817

19808367766356367096522727921316089451547342396490948067779940625178

88020116160602011047647958441543061184800996681742861951458927608369

31921303463436907590593465227992579980690076538677526802642563241223

58778978568413308707865022089920596975426734290393003094530833538477

51070253043054985292670186337562849238518822912544387932065661784941

62666108221075583052234535354001732258294144569659354587932951541940

4998441803274619168045467726087340720754974495397486708986

... which is 1 greater than the abscissa of the common logarithm of the number.

From Daniel Forgues, Jan 14 2012: (Start)

The length 1.2669... * 10^1757 (shown above) itself has 1758 digits!

The last 100 digits of Bernays's number are 67^257^729 mod 10^100 =

99736332723695669681470601458905407678415512345606116119671058321958

37693587043524719438498607119427. (End)

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 2223 digits from Lewis Mammel)

P. Bernays, Sur le platonisme dans les mathématiques, L'Enseignement Mathématique 34 (1935), pp. 52-69. (See p. 61.)

P. Bernays, Platonism in mathematics, translated from the French by C. D. Parsons.

Efunda Engineering Fundamentals, Series Expansion of Exponential and Logarithmic Functions

K. Podnieks, Towards a real Finitism?

Formula

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.

Mathematica

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 *)

Prog

(bc) /* bc script for cygwin bash shell or other UNIX environment */
/* Explicitly scale to 4000 decimal places */
i=10^4000
/* Calculate natural log of 100/67 */
b=100*i/67
c=i*(b-i)/(b+i)
x=c
s=0
for( j=1 ; x/j >0 ; j=j+2 ){
s = s + x/j
x = x*c/i*c/i
j /* progress mark */
}
s=2*s
/* Now s is the natural log of 100/67 */
/* Calculate natural log of sqrt(sqrt(10)) */
b=sqrt(10*i*i)
b=sqrt(b*i)
c=i*(b-i)/(b+i)
x=c
t=0
for( j=1 ; x/j >0 ; j=j+2 ){
t = t + x/j
x = x*c/i*c/i
j /* progress mark */
}
t=2*t
/* Now t is the integer part of 10^4000 * ln sqrt(sqrt(10)) */
ln10=4*t
/* ln10 is the integer part of 10^4000 * ln 10 */
ln67 = 2*ln10 - s
/* ln67 is the integer part of 10^4000 * ln 67 */
lg67 = ln67*i/ln10
/* lg67 is the integer part of 10^4000 * log_10 67 */
a=257^729
lgb = a*lg67
/* lgb is the integer part of 10^4000 * log_10 67^257^729 */
absc = lgb/i
/* absc is the abscissa of lgb, and its value is one less than the
number of decimal digits in Bernays's number */
mant = lgb - i*absc
/* Find number of digits in abscissa */
x=absc
for( nab=0 ; x>0 ; nab++ ) x = x/10
/* reduce the scale by nab */
mant = mant/10^nab
ln10 = ln10/10^nab
i = i/10^nab
/* find ln 10^mant */
lnmant = mant*ln10/i
/* calculate exp(lnmant) to get leading digits of Bernays's number */
fac=1
x=i
n=0
for( j=0 ; x/fac > 0 ; j++ ){
n = n + x/fac
x=x*lnmant/i
fac = fac*(j+1)
j /* progress mark */
}
/* display abscissa of log_10 ( Bernays's number ) */
absc
/* Display leading digits of Bernays's number.
( Truncation is to avoid displaying round-off error )*/
n / 10^20

Crossrefs

Sequence in context: A342138 A029619 A049922 * A243786 A278743 A081974

Adjacent sequences: A160103 A160104 A160105 * A160107 A160108 A160109

Keyword

cons,nonn,fini

Author

Lewis Mammel (l_mammel(AT)att.net), May 02 2009

Status

approved