Archimedes' Count of Homer's Cattle of the Sun

In the first few lines of The Odyssey, Homer foretells how Odysseus' crew “perished through their own sheer folly in eating the cattle of the Sun-god Hyperion”. These Cattle of the Sun grazed near the Sicilian town of Taormina (Tauromenion to its ancient Greek settlers) and, although endlessly warned not to, Odysseus' crew slaughtered some of them for food. For this sacrilege Zeus tossed them from their ship to their deaths with his thunderbolts, leaving Odysseus to continue his odyssey alone. In describing the sacred cattle, Homer indirectly gives their count by writing that they comprised seven herds containing fifty cattle each (Book XII: “Of oxen fifty head in every herd feed, and their herds are seven”), leaving it to the reader to determine the total number of cattle. Centuries later this simple multiplication problem was the inspiration for Archimedes' famous “Cattle Problem”, whose first line is: “If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily”. Archimedes, who lived in the Sicilian-Greek city-state of Syracuse, 85 kilometers south of Taormina, would have been very familiar with Homer's tale. In his problem, Archimedes challenges his colleague Eratosthenes to compute the number of the Cattle of the Sun having a larger and more complicated composition than the one described by Homer. Archimedes' problem is so complicated that the total number of cattle contains 206,545 digits. In this article I describe the origins of this problem in antiquity, its rediscovery in the eighteenth century, and the attempts since then to solve it. Its complete resolution had to await the computer age, since before then someone estimated that it would take the work of “a thousand men for a thousand years” to determine the exact solution. Attempts at its solution fueled the field of Diophantine Analysis — the analysis of problems whose solutions are restricted to whole numbers — and, in particular, the study of the so-called Pell Equation. Today a notebook computer using sophisticated algorithms can generate the number of cattle in seconds, taking more time to print out the number than to actually compute it. The amount of intellectual activity that has surrounded this problem over 23 centuries suggests the validity of Voltaire's remark, “There was more imagination in the head of Archimedes than in that of Homer”.