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Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773.[1]
The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by A. Amthor. Using logarithmic tables, he calculated the first digits of the smallest solution, showing that it is about 7.76Ã10206544{displaystyle 7.76times 10^{206544}} cattle, far more than could fit in the observable universe.[2] The decimal form is too long for humans to calculate exactly, but multiple precision arithmetic packages on computers can write it out explicitly.
![]() History[edit]
In 1769, Gotthold Ephraim Lessing was appointed librarian of the Herzog August Library in Wolfenbüttel, Germany, which contained many Greek and Latin manuscripts.[3] A few years later, Lessing published translations of some of the manuscripts with commentaries. Among them was a Greek poem of forty-four lines, containing an arithmetical problem which asks the reader to find the number of cattle in the herd of the god of the sun. It is nowgenerally credited to Archimedes.[4][5]
Problem[edit]
The problem, from an abridgement of the German translations published by Georg Nesselmann in 1842, and by Krumbiegel in 1880, states:
Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to color into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows:
If thou canst give, O friend, the number of each kind of bulls and cows, thou art no novice in numbers, yet can not be regarded as of high skill. Consider, however, the following additional relations between the bulls of the sun:
If thou hast computed these also, O friend, and found the total number of cattle, then exult as a conqueror, for thou hast proved thyself most skilled in numbers.[6]
Solution[edit]
The first part of the problem can be solved readily by setting up a system of equations. If the number of white, black, dappled, and yellow bulls are written as W,B,D,{displaystyle W,B,D,} and Y{displaystyle Y}, and the number of white, black, dappled, and yellow cows are written as w,b,d,{displaystyle w,b,d,} and y{displaystyle y}, the problem is simply to find a solution to:
which is a system of seven equations with eight unknowns. It is indeterminate, and has infinitely many solutions. The least positive integers satisfying the seven equations are:
which is a total of 50,389,082 cattle[6] and the other solutions are integral multiples of these. Note that the first four numbers are multiples of 4657, a value which will appear repeatedly below.
The general solution to the second part of the problem was first found by A. Amthor[7] in 1880. The following version of it was described by H. W. Lenstra,[2] based on Pell's equation: the solution given above for the first part of the problem should be multiplied by
where
and j is any positive integer. Equivalently, squaring w results in,
where {u, v} are the fundamental solutions of the Pell equation
The size of the smallest herd that could satisfy both the first and second parts of the problem is then given by j = 1, and is about 7.76Ã10206544{displaystyle 7.76times 10^{206544}} (first solved by Amthor). Modern computers can easily print out all digits of the answer. This was first done at the University of Waterloo, in 1965 by Hugh C. Williams, R. A. German, and Charles Robert Zarnke. They used a combination of the IBM 7040 and IBM 1620 computers.[8]
Pell equation[edit]
The constraints of the second part of the problem are straightforward and the actual Pell equation that needs to be solved can easily be given. First, it asks that B+W should be a square, or using the values given above,
thus one should set k = (3)(11)(29)(4657)q2 for some integer q. That solves the first condition. For the second, it requires that D+Y should be a triangular number,
Solving for t,
Substituting the value of D+Y and k and finding a value of q2 such that the discriminant of this quadratic is a perfect square p2 entails solving the Pell equation,
Amthor's approach discussed in the previous section was essentially to find the smallest v such that it is integrally divisible by 2*4657. The fundamental solution of this equation has more than 100,000 digits.
References[edit]
Further reading[edit]
External links[edit]
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Archimedes%27s_cattle_problem&oldid=927907820'
Quote fromIn other news, the server's gone through a major overhaul. I have added ranks to the server, which can be purchased using Belly gained from killing mobs.And as some of you may or may not have noticed, there is a new world added to the server. This world's oceans are approximately 3x as wide, giving it a more open feel.
I have also removed pirate ships from spawning in the world, as devil fruits can now be obtained searching via /treasure command. More information can be found at the new /spawn.To those of you worrying about your inventories, don't worry. They are safe (albeit locked away in a map you probably can't access ). I will let you bring over all of your items from the original world, excluding devil fruits. Doriki, class, and race cannot be transferred to the new map so those of you who have already gotten ahead are going to have to start over.
Sorry but there is no work around for this.I hope you guys enjoy the new map, I worked really hard getting it ready. If you have any problems with anything don't hesitate to PM me for help. Just want to say thank you to everyone for being so patient while I've been working on some last minute changes. The feedback I've gotten has been tremendous and I am proud to say that we have disabled our whitelist. Thanks to some help from scarydreamer I was able to get Cauldron running and we are even running some grief protection plugins, thus I feel comfortable enough opening to the public.
If you need help in-game you can use /help (or talk to me), hopefully it'll be smooth sailing from here on out.tl;dr bump. Quote fromJust want to say thank you to everyone for being so patient while I've been working on some last minute changes. The feedback I've gotten has been tremendous and I am proud to say that we have disabled our whitelist. Thanks to some help from scarydreamer I was able to get Cauldron running and we are even running some grief protection plugins, thus I feel comfortable enough opening to the public. If you need help in-game you can use /help (or talk to me), hopefully it'll be smooth sailing from here on out.tl;dr bumpDepending how things play off this summer I may be able to get on the server (it'll most likely be at night) but I should still be able to help with things like buildings. Quote fromIt became automatically private when I placed it down, but I guess the switch to essentials changed that. Either way, its still a problem.
Someone went out of their way to steal everything that I own that wasn't in my inventory.Sorry that happened to you. I'll sort something out when I see you online.In other news, the server's gone through a major overhaul. I have added ranks to the server, which can be purchased using Belly gained from killing mobs.And as some of you may or may not have noticed, there is a new world added to the server. This world's oceans are approximately 3x as wide, giving it a more open feel.
I have also removed pirate ships from spawning in the world, as devil fruits can now be obtained searching via /treasure command. More information can be found at the new /spawn.To those of you worrying about your inventories, don't worry. They are safe (albeit locked away in a map you probably can't access ). I will let you bring over all of your items from the original world, excluding devil fruits. Doriki, class, and race cannot be transferred to the new map so those of you who have already gotten ahead are going to have to start over.
Sorry but there is no work around for this.I hope you guys enjoy the new map, I worked really hard getting it ready. If you have any problems with anything don't hesitate to PM me for help.
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