关于乘方求余的定理
Theorems on residues obtained by the division of powers
摘要：
这是欧拉的“Theoremata circa residua ex divisione potestatum relicta”的英文译本，Novi Commentarii Muslimiae scientiarum Petropolitarum 7（1761），49-82。Enestrom指数中的E262。欧拉给出了关于素数p的幂剩余的许多初等结果，他证明了F_p^* 中由元素a生成的子群的阶必须除F_p^* 的阶p-1（即拉格朗日定理对于循环群的一个特例）。欧拉还证明了费马的小定理，即对于p的互素，a^{p-1} = 1 mod p（即不是0 mod p）。他说这个证明更自然，因为它使用了F_p^* 的乘法性质而不是二项式展开。感谢Jean-Marie Bois指出一些错别字。
英文摘要：
This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results on power residues modulo a prime number p. He shows that the order of a subgroup generated by an element a in F_p^* must divide the order p-1 of F_p^* (i.e. a special case of Lagrange's theorem for cyclic groups). Euler also gives a proof of Fermat's little theorem, that a^{p-1} = 1 mod p for a relatively prime to p (i.e. not 0 mod p). He remarks that this proof is more natural, as it uses multiplicative properties of F_p^* instead of the binomial expansion. Thanks to Jean-Marie Bois for pointing out some typos.
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