Talk:Archimedes's cattle problem

Untitled

"It is not known how the sun god managed to fit his herd into Sicily." Cracked me up! -80.57.138.148 19:03, 6 February 2007 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Archimedes's cattle problem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Black herd = (1/3 + 1/4) * white cows,

isn't the same as:

w = (7/12) * (B + b)

right?

Last edited at 18:40, 30 December 2010 (UTC). Substituted at 01:46, 5 May 2016 (UTC)

WP:NOTPAPER

... so let's give the 100K digits. David Eppstein? E Eng 03:39, 27 December 2017 (UTC)[reply]

Ok, done. Unless you meant, type them all in here? —David Eppstein (talk) 05:41, 27 December 2017 (UTC)[reply]

Well, I didn't mean by hand. E Eng 05:44, 27 December 2017 (UTC)[reply]

I'm not sure why we would want to supply the 206,545 digits of the smallest solution when it is just one of an infinite family of solutions. What's important is that the methodology in the Solution section is self-contained and reliable. Towards that end, I checked to see if the very long digits in the solution are correct, so solving for the smallest total number of cattle, T, I have the exact integer solution in Mathematica. Cross-checking the Lenstra methodology with the 1965 calculation that took 7 hours 49 minutes for Williams, German and Zarnke takes but less than two seconds using https://develop.open.wolframcloud.com/app/ . Here I use a bit of algebraic number theory to represent the w of the Solution section in a finite extension of the field of rationals, enabling me to easily prove that k[1] is indeed an exact integer, and consequently so is T.

dPrime = 609 * 7766 ;
d = dPrime * (2 * 4657)^2  ;
{root609, root7766, rootDPrime, w} = ToNumberField[{
    Sqrt[609], Sqrt[7766], Sqrt[dPrime],
    300426607914281713365 Sqrt[609] + 84129507677858393258 Sqrt[7766]
}, All] ;
k[j_] := (w^( 4658 j) - w^(-4658 j ))^2 / ( 4657 * 79072 ) ;
T = 50389082 k[1]

But, of course this method is sensitive to typos, like most math, which is why I wanted to double-check it. Dakleman (talk) 07:39, 12 February 2019 (UTC)[reply]

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Trivial solution

Wouldn't a trivial solution to this problem be that there are simply no cattle whatsoever? If Archimedes believed there was a solution, it was almost certainly this, as he could not possibly have predicted the invention of the computers which were required to find the smallest non-zero solution. This also satisfies the "smallest solution" requirement. Lurlock (talk) 16:26, 24 November 2019 (UTC)[reply]

My understanding of the ancient Greek relationship to the number 0 (see 0#Classical_antiquity) is that they would not have understood the idea that 0 is an answer to the question. --JBL (talk) 16:47, 24 November 2019 (UTC)[reply]

Hmm. Still, since 0 is arguably a valid solution for this problem, it should probably be mentioned in the article, perhaps with a disclaimer stating that this is not what he had in mind if that's the case. Lurlock (talk) 04:35, 25 November 2019 (UTC)[reply]

Wait, I just realized why this won't work: The number of bulls is greater than the number of cows. That would unfortunately require there to be a non-zero number of cattle... Lurlock (talk) 14:23, 25 January 2020 (UTC)[reply]