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This question already has an answer here:

Who first distinguished number theory and numerology?

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marked as duplicate by Geremia, Alexandre Eremenko, José Carlos Santos, Community Oct 3 at 15:00

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  • $\begingroup$ I think it is generally helpful to post more than one sentence. The term "numerology" only appeared in the early 1900s (and can refer to some traditional pursuits as well as loosely to pseudoscience). "Number theory" is also stretchable, the more so the further back we go. So it would help to specify what exactly is meant by each, and what counts as "distinguishing" prior to Hara's 1907 book. $\endgroup$ – Conifold Oct 3 at 0:51
  • $\begingroup$ There was never any confusion between number theory and numerology. $\endgroup$ – Alexandre Eremenko Oct 3 at 3:01
  • $\begingroup$ We do not know... IMO, at least for the Western world, there was the Pythagorean "discovery" of the properties of numbers, from which began number theoy; only later began numerology. $\endgroup$ – Mauro ALLEGRANZA Oct 3 at 6:02
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Looking for methods is theory, looking for results is... whatever. The interest in pythagorean triples or magic squares seems to have evolved from knowing some items to knowing a method to generate all of them. But that does not preclude a motivation e.g. some kind of power, magical or other, to be derived: knowledge is power.

A similar question aks when astronomy separated from astrology and there is no good answer. According to Westman Copernic was still looking for a better astrology, but in his approach he was following Ptolemy who had already separated the method, that is astronomy, from the application - astrology.

It seems obvious that Euclid's proof about the infinity of primes is number theory, so at least this a terminus ante quem. Ever since Burkert's book the view about so called pythagoreans has radically changed and currently Zhmud has worked out details various writings. Also one should remember that little is known about ancient Greek isopsephy which became popular at the same time when neopythagoreanism was in vogue.

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  • $\begingroup$ What's "isopsephy"? $\endgroup$ – Geremia Oct 4 at 23:00

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