1,1

The number has 206545 digits. Archimede's cattle problem, in equation form, requires the smallest sum W+X+Y+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 one hundred digits are 0303265435652072678728835138492561669543896048155005994630144292500354883118973723406626719455081800. - Robert G. Wilson v Sep 02 2004. See link below.

D. Barthe, "Le probleme des boeufs du Soleil", Les equations algebriques, pp. 134-9 Tangente Hors serie 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, 'Archimede's 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.

D. Olivastro, Ancient Puzzles, "Archimedes Revenge", pp. 184-7, Bantam Books NY 1993.

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. 112-3 April 2000.

I. Vardi, "Archimede's Cattle Problem", Amer. Math. Month. Vol. 105(4) April 1998 pp. 305-319, MAA Washington DC.

A. Weil, Number Theory, An approach through history from Hammurapi to Legendre, pp. 18-19, Birkhaeuser Boston 2001.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 187 (Entry 4729494) Penguin Books 1987.

Anonymous, The Archimedian Cattle Problem

E. Brown, Three Connections to Continued Fractions:Archimedes and the Cattle (pages 6-7/12)

B. Carroll, Archimeses 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

A. Veling, Full Solution Printout

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Robert G. Wilson v, Complete decimal expansion of the number

A. Winans, Archimedes' Cattle Problem and Pell's Equation

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] (from Robert G. Wilson v Sep 02 2004 using A. Winans's formula)

Adjacent sequences: A096148 A096149 A096150 this_sequence A096152 A096153 A096154

Sequence in context: A021932 A065472 A081112 this_sequence A021567 A019619 A098592

fini,nonn

Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 27 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 30 2004