I was reading Whiteside's article called "Newton the Mathemtician", where he says that Newton did Number Theory (e.g. inverstigating which numbers are expressible as a sum of two cubes). If this is true, can someone point me towards some papers of newton on number theory? If not, then do you know which book in the eight-volume Newton's mathematical papers I can find his number theory work? I'm very interested in this, it came as a huge surprise (perhaps it shouldn't have) to me that Newton even knew of number theory.

Concrete questions:

  1. Where can I find Newton's number theoeretical work in Whiteside's "Newton's Mathematical Papers" volumes?
  2. What did Newton investigate in Number Theory, and what did he discover?


EDIT: Here is the extract from Whiteside's article where he talks about Newton doing NT: enter image description here

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    $\begingroup$ Why don't you look up the footnotes in that paragraph? They might tell you the sources of the remarks they are attached to. $\endgroup$ – KCd Oct 1 '14 at 23:47
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    $\begingroup$ Look at Newton's Arithmetica Universalis. In Bashmakova and Smirnova's "The Beginnings and Evolution of Algebra" (pp. 137-138) they write that in that work Newton determined the ring of integers for a real quadratic field, which would be called number theory today. The same remark is made in a footnote on p. 106 of "Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory." $\endgroup$ – KCd Oct 2 '14 at 12:29
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    $\begingroup$ If you look in the author indices in each volume of Dickson's "History of the Theory of Numbers", he mentions Newton quite rarely. $\endgroup$ – KCd Oct 2 '14 at 12:32
  • $\begingroup$ Just google "Newton mathematical papers arithmetic universalis" and you can answer that yourself. $\endgroup$ – KCd Oct 3 '14 at 12:22

I would have written this in a comment but I don't have the rep yet.

In Stillwell's beautiful article about elliptic curves (in the also great book Mathematical Evolutions) he cites a paper where Newton recognizes Diophantus parametrization of the cubic $x^3-3x^2+3x+1=y^2$ as a use of geometry to get results in number theory (you can find the bibliographical reference in the last page of the linked article)


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