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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 with
algebra, including the
algebraic equations now known as Diophantine equations. One example is the equation,
x4 + y4 + z4 = w4, which can be immediately identified as unusual, since we resort to using a
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
958004 + 2175194 + 4145604 = 4224814.
A far simpler Diophantine equation is the
cubic equation,
x3 + w3 = y3 + z3, 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:
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, x3 + w3 = y3 + z3 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 1019. (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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