  Translated from the Latin original, "Theorema arithmeticum eiusque
demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in
the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper
evaluates \[
\frac{a^n}{(a-b)(a-c)(a-d)...(a-x)}+\frac{b^n}{(b-a)(b-c)(b-d)...(b-x)}
+...+\frac{x^n}{(x-a)(x-b)(x-c)...(x-\upsilon)}. \] When $n \leq m-2$, the sum
is 0, which Euler had already shown in sect. 1169 of his Institutiones calculi
integralis, vol. I, E366. When $n=m-1$ the sum is $=1$. When $n \geq m$, then
using Newton's identities Euler gets expressions in terms of sums of powers of
the numbers a,b,c,d,...,\upsilon,x.
  The notation used in this paper is confus(ing/ed), and indeed the editors of
the Opera omnia made several corrections to the original publication. Near the
end of the paper, "... for each of them one searches for what the character
will be..." is the only reading of the sentence I can come up with.
  Examples of a work that refers to this theorem is J.-P. Serre, Corps locaux,
2nd ed., ch. III, sect. 6 (for $n \leq m-1$), and D. E. Knuth, The Art of
Computer Programming, vol. 1, sect. 1.2.3, exercise 33.
