By de Weger B.M.M.
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Additional info for Algorithms for Diophantine Equations [PhD Thesis]
596–ca. 475 BCE) and the Pythagoreans, who were members of a school that was active in the city of Crotona, in 32 | NUMBERS AND MAGNITUDES IN THE GREEK TRADITION what is today the south of Italy. Many legends and myths came to be associated with their name, and it is sometimes diﬃcult to separate such legends from historical truth. Nevertheless, there is no doubt about the seminal importance of their mathematical contributions. The best known of these is the famous theorem about right-angled triangles that bears the name of Pythagoras: the square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides.
But we can easily think of the following alternative ordering: 1, 3, 5, 7, . . 2, 4, 6, 8, . . , 80 is “greater than” 8,000,003). Notice that this is really a diﬀerent ordering and not just a way of renaming the members of the sequence. How do we see this? , 1), whereas in the alternative order just presented, there are two elements with that property, namely, 1 and 2. Indeed, in the alternative ordering, 2 is greater than any given odd number, but no speciﬁc odd number can be said to appear immediately before the number 2.
Our interest in the Pythagoreans, at any rate, lies in a diﬀerent aspect of their work, namely, the focused attention and intense research they devoted to natural numbers and their properties. The interest of the Pythagoreans in numbers goes way beyond the purely arithmetical. For them, number was the universal principle that underlies the cosmos and allows it to be understood. As part of a unique blend of a rational approach to understanding nature with numerology and other mystical practices, the Pythagoreans saw the natural numbers as a clearly discernible, stable element that hides behind the apparent chaos of day-to-day experience and helps to make sense of it.
Algorithms for Diophantine Equations [PhD Thesis] by de Weger B.M.M.