A096151 Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem.

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, 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, 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

Offset: 206545

Author

Lekraj Beedassy, Jul 27 2004

Keywords

• cons
• fini
• full
• nonn

Comments

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.

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]

The final 100 digits are 0303265435652072678728835 1384925616695438960481550 0599463014429250035488311 8973723406626719455081800. - Robert G. Wilson v, Sep 02 2004. [See link below.]

References

• A. Amthor, "Das Problema bovinum des Archimedes", Zeitschrift f. Math. u. Physik (Hist.-litt.Abtheilung), Vol. XXV (1880), pp 153-171.
• D. Barthe, "Le problème des boeufs du Soleil", Les équations algébriques, pp. 134-9 Tangente Hors série No. 22 Pole Paris 2005.
• A. H. Beiler, Recreations in the Theory of Numbers, pp. 249-251, Dover NY 1966.
• E. T. Bell, The Last Problem, pp. 148-152, MAA Washington DC 1990.
• K. Devlin, All The Math That's Fit To Print, pp. 64, MAA Washington DC 1994.
• L. E. Dickson, History of the Theory of Numbers, Vol.II, pp. 342-5, Chelsea NY 1992.
• H. Doerrie, 100 Great Problems of Elementary Mathematics, Prob.1, "Archimedes' Problema Bovinum", pp. 3-7 Dover NY 1965.
• A. P. Domoryad, Mathematical Games and Pastimes, pp. 29-30 Pergamon Press NY 1963.
• P. Haber, Mathematical Puzzles and Pastimes, Prob. 113, pp. 40-1; 60-3, The Peter Pauper Press NY 1957.
• P. Hoffman, Archimedes' Revenge, pp. 29-32 Penguin 1988.
• M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. v-vi.
• H. L. Nelson, "A Solution to Archimedes' Cattle Problem", Journal of Recreational Mathematics, Vol. 13:3 (1980-81), pp. 162-176.
• D. Olivastro, Ancient Puzzles, "Archimedes Revenge", pp. 184-7, Bantam Books NY 1993.
• M. Petkovic, "Archimedes Cattle Problem", Famous Puzzles of Great Mathematicians, pp. 41-3, Amer. Math. Soc.(AMS), Providence RI 2009.
• W. L. Schaaf, Recreational Mathematics: A Guide To Literature, p. 31, NCTM Washington DC 1963.
• A. Weil, Number Theory, An approach through history from Hammurapi to Legendre, pp. 18-19, Birkhäuser Boston 2001.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 187 (Entry 4729494) Penguin Books 1987.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 206545..413089
• Robert G. Wilson v, Complete decimal expansion of the number (complete sequence, but not in b-file format).
• Anonymous, The Archimedian Cattle Problem
• Alex Bellos and Brady Haran, The Archimedes Number Numberphile video (2019)
• E. Brown, Three Connections to Continued Fractions:Archimedes and the Cattle (pages 6-7/12)
• B. Carroll, Archimedes and Large Numbers: Cattle Puzzle
• Code Golf and Coding Challenges Challenge, Archimedes's cattle problem
• K. Devlin, The Archimedes Cattle Problem
• H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
• I. Peterson, Mathtrek, Cattle of the Sun
• T. Rike, Archimedes Cattle Problem
• C. Rorres, The Cattle Problem
• Ian Stewart, Counting the Cattle of the Sun, Mathematical Recreations Column, Scientific American pp. 112, Apr 03 2000.
• Ilan Vardi, Archimedes' Cattle Problem, Amer. Math. Month. Vol. 105(4) April 1998 pp. 305-319, MAA Washington DC.
• A. Veling, Solution To Archimedes' Cattle Problem (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
• A. Veling, Full Solution Printout (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
• Eric Weisstein's World of Mathematics, Archimedes' Cattle Problem
• H. C. Williams, R. A. German, and C. R. Zarnke, Solution of the cattle problem of Archimedes, Mathematics of Computation, Vol. XIX (1965), pp. 671-687.
• A. Winans, Archimedes' Cattle Problem and Pell's Equation

Crossrefs

See A003131 for another example of a sequence with a large offset based on a large integer. - N. J. A. Sloane, Dec 25 2018

Programs

Mathematica

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 *)

Extensions

More terms from Robert G. Wilson v, Jul 30 2004

Reference added and two links fixed by William Rex Marshall, Nov 17 2010

Edited (broken links fixed, historical references added) by M. F. Hasler, Feb 13 2013

Offset corrected by N. J. A. Sloane, Dec 25 2018