

A096151


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


3



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
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OFFSET

206545,1


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 50389082*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).
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 153171.
D. Barthe, "Le probleme des boeufs du Soleil", Les equations algebriques, pp. 1349 Tangente Hors serie No. 22 Pole Paris 2005.
A. H. Beiler, Recreations in the Theory of Numbers, pp. 249251, Dover NY 1966.
E. T. Bell, The Last Problem, pp. 148152, 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. 3425, Chelsea NY 1992.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Prob.1, "Archimedes' Problema Bovinum", pp. 37 Dover NY 1965.
A. P. Domoryad, Mathematical Games and Pastimes, pp. 2930 Pergamon Press NY 1963.
P. Haber, Mathematical Puzzles and Pastimes, Prob. 113, pp. 401; 603, The Peter Pauper Press NY 1957.
P. Hoffman, Archimedes' Revenge, pp. 2932 Penguin 1988.
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. vvi.
H. L. Nelson, "A Solution to Archimedes' Cattle Problem", Journal of Recreational Mathematics, Vol. 13:3 (198081), pp. 162176.
D. Olivastro, Ancient Puzzles, "Archimedes Revenge", pp. 1847, Bantam Books NY 1993.
M. Petkovic, "Archimedes Cattle Problem", Famous Puzzles of Great Mathematicians, pp. 413, Amer. Math. Soc.(AMS), Providence RI 2009.
W. L. Schaaf, Recreational Mathematics: A Guide To Literature, p. 31, NCTM Washington DC 1963.
I. Stewart, "Counting the Cattle of the Sun" in Mathematical Recreations Column, Scientific American pp. 112, Apr 03 2000.
I. Vardi, "Archimedes' Cattle Problem", Amer. Math. Month. Vol. 105(4) April 1998 pp. 305319, MAA Washington DC.
A. Weil, Number Theory, An approach through history from Hammurapi to Legendre, pp. 1819, BirkhĂ¤user Boston 2001.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 187 (Entry 4729494) Penguin Books 1987.
H. C. Williams, R. A. German, and C. R. Zarnke, "Solution of the cattle problem of Archimedes", Mathematics of Computation, Vol. XIX (1965), pp. 671687.


LINKS

Table of n, a(n) for n=206545..206649.
Robert G. Wilson v, Complete decimal expansion of the number (complete sequence, but not in bfile format).
Anonymous, The Archimedian Cattle Problem
E. Brown, Three Connections to Continued Fractions:Archimedes and the Cattle (pages 67/12)
B. Carroll, Archimedes and Large Numbers: Cattle Puzzle
K. Devlin, The Archimedes Cattle Problem
I. Peterson, Mathtrek, Cattle of the Sun
T. Rike, Archimedes Cattle Problem
C. Rorres, The Cattle Problem
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
A. Winans, Archimedes' Cattle Problem and Pell's Equation


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 *)


CROSSREFS

See A003131 for a another example of a sequence with a large offset based on a large integer.  N. J. A. Sloane, Dec 25 2018
Sequence in context: A244675 A065472 A081112 * A021567 A019619 A177436
Adjacent sequences: A096148 A096149 A096150 * A096152 A096153 A096154


KEYWORD

cons,fini,nonn


AUTHOR

Lekraj Beedassy, Jul 27 2004


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


STATUS

approved



