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I would regard as 'elementary number theory' that what needs no previous mathematical knowledge, esp. no abstract algebra, no Galois theory and no (complex) analysis. 'Elementary number theory' could include what Euler and Gauss did and of course "elementary" means not "simple". I would regard the reducability of statements like the four colour theorem to diophantine ones as belonging to elementary number theory too. BTW, if I remember correctly, Euler wrote an algebra textbook containing the Fermat for exp.=3 case with the help of an uneducated pupil to guarantee it's elementary nature.