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%I #87 Aug 05 2022 07:46:22
%S 7,7,6,0,2,7,1,4,0,6,4,8,6,8,1,8,2,6,9,5,3,0,2,3,2,8,3,3,2,1,3,8,8,6,
%T 6,6,4,2,3,2,3,2,2,4,0,5,9,2,3,3,7,6,1,0,3,1,5,0,6,1,9,2,2,6,9,0,3,2,
%U 1,5,9,3,0,6,1,4,0,6,9,5,3,1,9,4,3,4,8,9,5,5,3,2,3,8,3,3,0,3,3,2,3,8,5,8,0
%N Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem.
%C The number has 206545 digits. Archimedes's cattle problem, in equation form, requires the smallest sum W+X+Y+Z+w+x+y+z of the system W = (1/2 + 1/3)*X + Z; X = (1/4 + 1/5)*Y + Z; Y = (1/6 + 1/7)*W + Z; w = (1/3 + 1/4)*(X+x); x = (1/4 + 1/5)*(Y+y); y = (1/5 + 1/6)*(Z+z); z = (1/6 + 1/7)*(W+w), subject to the conditions that W+X be a square and Y+Z be triangular.
%C This in turn reduces to computing the value 224571490814418*t(1)^2, where (s(1), t(1)) is the smallest nontrivial solution to s^2 - D*t^2 = 1, with D=410286423278424 (or smallest solution t divisible by 9314 for squarefree D=4729494). [First number changed resulting from answers to a code golf challenge regarding this sequence by _Jonathan Oswald_, Jun 25 2020]
%C The final 100 digits are 0303265435652072678728835 1384925616695438960481550 0599463014429250035488311 8973723406626719455081800. - _Robert G. Wilson v_, Sep 02 2004. [See link below.]
%D A. Amthor, "Das Problema bovinum des Archimedes", Zeitschrift f. Math. u. Physik (Hist.-litt.Abtheilung), Vol. XXV (1880), pp 153-171.
%D D. Barthe, "Le problème des boeufs du Soleil", Les équations algébriques, pp. 134-9 Tangente Hors série No. 22 Pole Paris 2005.
%D A. H. Beiler, Recreations in the Theory of Numbers, pp. 249-251, Dover NY 1966.
%D E. T. Bell, The Last Problem, pp. 148-152, MAA Washington DC 1990.
%D K. Devlin, All The Math That's Fit To Print, pp. 64, MAA Washington DC 1994.
%D L. E. Dickson, History of the Theory of Numbers, Vol.II, pp. 342-5, Chelsea NY 1992.
%D H. Doerrie, 100 Great Problems of Elementary Mathematics, Prob.1, "Archimedes' Problema Bovinum", pp. 3-7 Dover NY 1965.
%D A. P. Domoryad, Mathematical Games and Pastimes, pp. 29-30 Pergamon Press NY 1963.
%D P. Haber, Mathematical Puzzles and Pastimes, Prob. 113, pp. 40-1; 60-3, The Peter Pauper Press NY 1957.
%D P. Hoffman, Archimedes' Revenge, pp. 29-32 Penguin 1988.
%D M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. v-vi.
%D H. L. Nelson, "A Solution to Archimedes' Cattle Problem", Journal of Recreational Mathematics, Vol. 13:3 (1980-81), pp. 162-176.
%D D. Olivastro, Ancient Puzzles, "Archimedes Revenge", pp. 184-7, Bantam Books NY 1993.
%D M. Petkovic, "Archimedes Cattle Problem", Famous Puzzles of Great Mathematicians, pp. 41-3, Amer. Math. Soc.(AMS), Providence RI 2009.
%D W. L. Schaaf, Recreational Mathematics: A Guide To Literature, p. 31, NCTM Washington DC 1963.
%D A. Weil, Number Theory, An approach through history from Hammurapi to Legendre, pp. 18-19, Birkhäuser Boston 2001.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 187 (Entry 4729494) Penguin Books 1987.
%H Robert G. Wilson v, <a href="/A096151/b096151.txt">Table of n, a(n) for n = 206545..413089</a>
%H Robert G. Wilson v, <a href="/A096151/a096151.txt">Complete decimal expansion of the number</a> (complete sequence, but not in b-file format).
%H Anonymous, <a href="http://www-gap.dcs.st-and.ac.uk/~history/Diagrams/ArchimedesCPSoln.gif">The Archimedian Cattle Problem</a>
%H Alex Bellos and Brady Haran, <a href="https://www.youtube.com/watch?v=dNxyFtqcNss">The Archimedes Number</a> Numberphile video (2019)
%H E. Brown, <a href="http://www.math.vt.edu/people/brown/doc/cfrax.pdf">Three Connections to Continued Fractions:Archimedes and the Cattle (pages 6-7/12)</a>
%H B. Carroll, <a href="http://departments.weber.edu/physics/carroll/Archimedes/numbers.htm">Archimedes and Large Numbers: Cattle Puzzle</a>
%H Code Golf and Coding Challenges Challenge, <a href="https://codegolf.stackexchange.com/questions/206456/archimedess-cattle-problem">Archimedes's cattle problem</a>
%H K. Devlin, <a href="http://www.maa.org/devlin/devlin_02_04.html">The Archimedes Cattle Problem</a>
%H H. W. Lenstra Jr., <a href="http://www.ams.org/notices/200202/fea-lenstra.pdf">Solving the Pell Equation</a>, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
%H I. Peterson, Mathtrek, <a href="http://www.maa.org/mathland/mathtrek_4_20_98.html">Cattle of the Sun</a>
%H T. Rike, <a href="http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/archimedes.pdf">Archimedes Cattle Problem</a>
%H C. Rorres, <a href="http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html">The Cattle Problem</a>
%H Ian Stewart, <a href="https://www.jstor.org/stable/26058681">Counting the Cattle of the Sun</a>, Mathematical Recreations Column, Scientific American pp. 112, Apr 03 2000.
%H Ilan Vardi, <a href="https://www.jstor.org/stable/2589706">Archimedes' Cattle Problem</a>, Amer. Math. Month. Vol. 105(4) April 1998 pp. 305-319, MAA Washington DC.
%H A. Veling, <a href="http://web.archive.org/http://www.veling.nl/anne/templars/Solution.html">Solution To Archimedes' Cattle Problem</a> (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
%H A. Veling, <a href="http://web.archive.org/http://veling.nl/anne/templars/archim.htm">Full Solution Printout</a> (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArchimedesCattleProblem.html">Archimedes' Cattle Problem</a>
%H H. C. Williams, R. A. German, and C. R. Zarnke, <a href="https://doi.org/10.1090/S0025-5718-65-99945-X">Solution of the cattle problem of Archimedes</a>, Mathematics of Computation, Vol. XIX (1965), pp. 671-687.
%H A. Winans, <a href="http://kobotis.net/math/UC/2000/projects/winans.pdf">Archimedes' Cattle Problem and Pell's Equation</a>
%t PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; x = 4729494; y = PellSolve[x]; z = Floor[25194541/184119152(y[[1]] + y[[2]]*Sqrt[x])^4658]; Take[ IntegerDigits[z], 105] (* _Robert G. Wilson v_, Sep 02 2004, using A. Winans's formula *)
%Y See A003131 for another example of a sequence with a large offset based on a large integer. - _N. J. A. Sloane_, Dec 25 2018
%K cons,fini,full,nonn
%O 206545,1
%A _Lekraj Beedassy_, Jul 27 2004
%E More terms from _Robert G. Wilson v_, Jul 30 2004
%E Reference added and two links fixed by _William Rex Marshall_, Nov 17 2010
%E Edited (broken links fixed, historical references added) by _M. F. Hasler_, Feb 13 2013
%E Offset corrected by _N. J. A. Sloane_, Dec 25 2018
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