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As a middle-school student, I encountered equations like

$$ x^2 - 92 y^2 = 1 .$$

But I remember that I was not interested or motivated at all. Why should I care about this kind of problem? For what kind of problems do I need to study it?

So, why did the ancients study it? Purely out of curiosity?

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  • $\begingroup$ This is called Pell equation, historical motivations include approximate extraction of square roots. $\endgroup$ – Conifold Mar 15 '17 at 4:14
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    $\begingroup$ Change the $1$ on the right to $-1$ and the real solutions qualitatively look the same: both are hyperbolas. But while $x^2 - 92y^2 = 1$ has infinitely many integer solutions, $x^2 - 92y^2 = -1$ has none. Is that difference not interesting? I don't mean this is why anyone studied the equations, but the drastic change in the integer solutions is an intriguing arithmetic phenomenon. $\endgroup$ – KCd Mar 17 '17 at 6:21
  • $\begingroup$ Diophantine Equations are intimately linked to Continued Fractions, which were of interest to the ancient Chinese but didn't become popular in the West until the 1700s or so. $\endgroup$ – Stella Biderman Mar 26 '17 at 0:07
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For the "ancients", "numbers" were integers or fractions. They did not have a concept of a real number. (They had an equivalent concept of length in geometry, but analytic geometry did not exist, and they did not think of geometric problems as problems about equations. Geometric problems had to be solved by construction). Interest to the theory of whole numbers in Greece can be traced back to Pythagoreans who defined the main objects (primes, squares, etc.) and started to study their properties. According to the tradition, Pythagoras claimed that "numbers rule the world" (he meant integers). This sweeping claim was based on his discovery that musical intervals correspond to certain fractions. (In the long run he was right as we know: the basic laws of nature have mathematical formulation).

The early stages of development of Diophantine equations are unknown to us. They were usually formulated as word problems. See, for example Archimedes' cattle problem on Wikipedia as one especially famous problem. It is about an equation of the same type you give, it is called Pell's equation nowadays. This is an isolated example, and we do not really know why exactly people were interested in such problems. Perhaps indeed out of curiosity. Most of the important mathematics was invented like this, by the way, out of curiosity.

Then, 3 centuries after Archimedes, we have the book of Diophantus which treats a lot of much more complicated problems like that. The modern development starts essentially with Fermat who read Diophantus...a millenium later. Nowadays this is one of the most fashionable areas of mathematics. There are some applications, but mostly it is driven by curiosity.

Ancient Babylonians also had taste to these problems, for example they knew how to find all solutions of $a^2+b^2=c^2$ in positive integers. Whether the Greeks started the subject independently or it came from Babylonia, it is not known.

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  • $\begingroup$ minor point of interest: the Greeks had ratios but not fractions. 1/2 is a ratio of two whole numbers, not the fracturing of one. by contrast, the Arabs had a concept of fraction, e.g.1/2 as a fragment of a unit. hence the term algebra, from al-jabr, mending of fractures (literally bone-setting). $\endgroup$ – mobileink Mar 20 '17 at 19:02
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One ancient mathematical topic that gives rise to Pell-type equations is polygonal numbers. Look at the list of triangular numbers (1, 3, 6, 19, 15, 21, 28, 36,...) and square numbers (1, 4, 9, 16, 25, 36, 49,...), and aside from the trivial common value 1 you will find another common value: 36. It is natural to ask if there are other "triangular-square numbers." There are in fact infinitely many, but the next example is greater than 1000.

Asking more generally if there are integers that are both $m$-gonal and $n$-gonal for different $m, n \geq 3$ leads to a Pell-type Diophantine equation.

I am not suggesting polygonal numbers are an important topic anymore, but the level of math the ancient mathematicians were thinking about could not be expected to be too high.

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