Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T15:19:37.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral solutions of ass and mule problems

Published online by Cambridge University Press:  01 August 2016

David Singmaster
David Singmaster*
Affiliation:
87 Rodenhurst Road, London SW4 8AF, e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In [1], Tomislav Došlić studies a special case of the classic Ass and Mule Problem and determines when integral data give integral solutions. The question of when integral data gives integral solutions in the general case intrigued me some years ago; I even published the assertion that I could not see any simpler result than computing the answers and seeing if the results were integral [2]. However, I later found a satisfactory result for this and some variants of the problem, which are given in [3, 4]. Here I give this result, which describes an infinite set, and show it leads to a simpler derivation of Došlić's interesting result that his variation has only 16 positive integral solutions. A different method permits finding the nonpositive integral solutions. I will then state a similar but unsolved problem.

Type
Articles
Information
The Mathematical Gazette , Volume 89 , Issue 516 , November 2005 , pp. 365 - 370
Copyright
Copyright © The Mathematical Association 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Došlić, Tomislav, Fibonacci in Hogwarts, Math. Gaz. 87 (November 2003) pp. 432–436.Google Scholar
2. Hadley, John and Singmaster, David, Problems to sharpen the young, Math. Gaz. 76 (March 1992) pp. 102–126. Problem 16.Google Scholar
3. Singmaster, David, Some diophantine recreations, in Berlekamp, Elwyn R. & Rodgers, Tom (eds.), The Mathemagician and Pied Puzzler A Collection in Tribute to Martin Gardner, Peters, A. K., (1999) pp. 219–235.Google Scholar
4. Singmaster, David, A variation of the Ass and Mule Problem, Crux Math. 28 [4] (May 2002) pp. 236–238.Google Scholar
5. Pisano, Leonardo, called Fibonacci, Liber Abaci, (1202; 2nd edn, 1228), in Boncompagni, B. (ed. and pub.), Scritti di Leonardo Pisano, vol. 1, (1857) pp. 190–203, 325–326, 332–333, 344–346.Google Scholar
5A. Sigler, Laurence E., Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation, (Springer) 2002.Google Scholar
6. Euclid, , Opera, c. 325 BC (edited by Heiberg, J. L. & Menge, H.) Teubner, Leipzig, 1916. Vol. 8, pp. 286–287.Google Scholar
7. Diophantus, , Arithmetica, c. 250, in Heath, T. L.; Diophantus of Alexandria (2nd ed.), OUP, 1910; reprinted by Dover, 1964. Book I, prob. 15, pp. 134–135. [See probs. 18 & 19, pp. 135–136, for extended versions.]Google Scholar
8. Metrodorus, (compiler), The Greek Anthology, c. 510. Translated by Paton, W. R., Loeb Classical Library, Harvard University Press, & Heinemann (1916–18) Vol. 5, Book 14, art. 145–146, p. 105.Google Scholar
9. Singmaster, David, Puzzles from the Greek Anthology, Math. Spectrum 17:1(1984–85) pp. 11–15.Google Scholar
10. Bhaskara (II) = Bhâskara, Bijaganita = Bîjaganita (1150) in Colebrooke, Henry Thomas (trans.) Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bháscara, John Murray, London (1817). (There have been several reprints, including Sändig, Wiesbaden, 1973.) Chap. IV, v. 106, p. 191.Google Scholar