### Diophantine Equations

July 29, 2024 As I wrote in a very early article (The Lonely Runner Conjecture, August 11, 2011), whenever I see a Diophantine equation, the word,*piebald*, comes to mind. This unusual word appears in the Archimedes' cattle problem, a Diophantine problem supposedly communicated by Archimedes to his friend, Eratosthenes.[1] You can view the original Greek text here.[2] It's a Diophantine equation system of seven equations in eight unknowns, but it can be solved with the requirement that the variables are positive natural numbers. The problem concerns the number of cattle owned by the Sun god, Helios, some of which were white, some blue, some yellow, and some piebald, that conform to certain restrictions. The smallest number solution, about 7.76x10

^{206544}cattle, was found by Carl Ernst August Amthor (1845–1916), headmaster of the Gymnasium of the Holy Cross in Dresden, Germany. The number of protons in the observable universe is about 10

^{80}.

*Piebald cow example. A Holstein Frisian cow on a pasture in the Rhön. Holsteins are the most common dairy cow in the United States.As a teenager, I would often assist my father in his construction business. We would sometimes travel to a rural area to give an cost estimate. One farm we visited had a sign at the entrance with an image of a cow and the words, "Registered Holsteins," below it.Rural folk know that Holstein is a breed of cattle. My father, however, was a city boy, and he addressed our potential customer as "Mrs. Holstein."(Wikimedia Commons image by Verum. Click for larger image.)*

Florian Cajori (1859-1930), whom I read as a student, uses the word, piebald,[1] but modern authors tend to use terms like dappled and spotted. According to the Google Ngram Viewer, the frequency of written occurrence for

*piebald*peaked in 1866 at 2.2257 x 10

^{-7}, dipped to 0.6185 x 10

^{-7}in 1984, and rebounded to 1.4610 x 10

^{-7}in 2019. That means you will only see this word after reading more than about 10 million words. Diophantus (c. 210 - c. 290) was a Greek mathematician who lived in Alexandria, Egypt. His major work,

*Arithmetica*, is concerned withalgebra, including the algebraic equations now known as Diophantine equations. One example is the equation,

**x**, which can be immediately identified as unusual, since we resort to using a

^{4}+ y^{4}+ z^{4}= w^{4}**w**variable to supplement the normally encountered

**x**,

**y**, and

**z**. Leonhard Euler (1707-1783) incorrectly conjectured that this equation has no nontrivial solutions, but it was later shown with computer assistance to have infinitely many nontrivial solutions. I sometimes do computational mathematics to sharpen my programming skill, but I'm happy I didn't address this equation, since the smallest nontrivial solution is

**95800**. A far simpler Diophantine equation is the cubic equation,

^{4}+ 217519^{4}+ 414560^{4}= 422481^{4}**x**, whose solutions are listed as sequence A001235 at the On-Line Encyclopedia of Integer Sequences. The smallest nontrivial solution of this equation is 1729, known as the taxicab number after a conversation between G. H. Hardy (1877 1947) and Srinivasa Ramanujan (1887-1920). There are infinitely many nontrivial solutions of this equation, and I was inspired to write a C language computer program to brute force the first few hundred. The source code for this program can be found here. My Linux Mint desktop computer with an Intel 13th Gen ten core i5-13400 processor and 16 gigabytes of memory found 648 solutions in less than three minutes, but it took a little more than seven minutes to find 938 solutions. The first fifty are as follow:

^{3}+ w^{3}= y^{3}+ z^{3}1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977, 805688, 842751, 885248, 886464, 920673, 955016, 984067, 994688, 1009736, 1016496, 1061424, 1073375, 1075032, 1080891, 1092728

*Nine hundred solutions of the Diophantine equation, x ^{3} + w^{3} = y^{3} + z^{3} found by my C language computer program. The solutions are not in ascending order in the data file, so you need to sort the data to get a plot like this. If you're willing to take some time, or have access to a much faster computer, a 64-bit compiler has a maximum value of an unsigned long integer of about 1.84 x 10^{19}. (Click for larger image.)*

### References:

- Florian Cajori, A History of Mathematics, MacMillan Company, (New York, 1894) , p. 73f. (via Project Gutenberg).
- Archimedes Cattle Problem, Greek text, New York University.
- Piebald, frequency of mentions from 1800-2019, with smoothing value of three, via Google Ngram Viewer.

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