A theorem of arithmetic and its proof
In this paper, Euler proves that for m unequal positive integers a, b, c, d, ..., the sum of the fractions: an/[(a–b)(a–c)(a–d)...] + bn/[(b–a)(b–c)(b–d)...] + cn/[(c–a)(c–b)(c–d)...] + dn/[(d–a)(d–b)(d–c)...} + ... is equal to 0 for nâ¤m–2, and he gives a general formula for the sum of these fractions for n=1, n=m and n>m. He shows a direct relationship between the values of the sum of these fractions for higher n and the coefficients of the polynomial (z–a)(z–b)(z–c)....
Original Source Citation
Commentationes arithmeticae collectae, Volume 2, pp. 588-592.
Opera Omnia Citation
Series 1, Volume 6, pp.486-493.