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"1+2+3+4=10" is an arithmetical triviality that is popular as an example of Pythagoreanism. There seems to be no mention of it in ancient Greek texts prior to Speusippus (e.g. ca. 350BCE; see Zhmud's Arithmology), so it seems rightly considered to be pseudo-pythagorean. There is however a possibility that it could have been noted and/or commented at earlier times in some other accessible culture, e.g. Egypt or India, etc (probably not Babylon where they used hexadecimal numbers). Refs to papers or sources mentioning such early occurences would be appreciated.

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    $\begingroup$ The problem with seeking references to this is that the references would provide essentially no information as to when someone "first" considered adding consecutive integers. This was likely done on many occasions thousands of years earlier than 350 BC, as surely such sums sometimes occurred in financial or trading transactions, as well as just playing around with counting stones or sticks. In fact, surely sums of the first few positive integers as corresponding to the number of objects in a triangular array dates back several thousand years BEFORE writing, probably over $10,000$ years ago. $\endgroup$ Commented Sep 27, 2020 at 21:34
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    $\begingroup$ I doubt that there are any. Even Herdmans, cranky authors that "uncover" secret Egyptian numerology everywhere have to admit that "The Pythagorean Mystery Numbers are not exactly what the Egyptians were up to", and the "Indian roots" of the tetractys are the invention of Blavatsky of the Theosophical Society, see Fitger, p.14. $\endgroup$
    – Conifold
    Commented Sep 28, 2020 at 6:45
  • $\begingroup$ @Cornifold. Thanks, I would have missed the Blavatsky connection: it is something like outfaking the fake. It seems however that Plato also has been also heading into cuckooland with his ideal ('eidetic') numbers. $\endgroup$
    – sand1
    Commented Sep 28, 2020 at 8:52
  • $\begingroup$ @DaveLRenfro. It is not a question 'Who first' but 'Did anybody think this triviality worth noting'. $\endgroup$
    – sand1
    Commented Sep 28, 2020 at 8:55
  • $\begingroup$ I still think various aspects of this identity would have been discovered/observed independently many times in prehistoric times from drawing triangular arrays of marks in sand/dirt or arranging objects in triangular arrays, perhaps observing that, at the 4th level, the number of marks/objects is equal to the number of fingers or perhaps otherwise trying to match the triangular numbers with numbers of things in other contexts. I think it's easy to underestimate just how many elementary observations/connections many tens of thousands of people each year can make over a period of 100,000+ years. $\endgroup$ Commented Sep 28, 2020 at 9:37

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