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On the Goldbach-Euler Theorem regarding Prime Numbers


IN the published correspondence of Euler there is a note from him to Goldbach, or, the other way, from Goldbach to Euler, in which a very wonderful theorem is stated which has never been proved by Euler or any one else, which I hope I may be able to do by an entirely improved method that I have applied with perfect success to the problem of partitions and to the more general problem of demonstration, i.e. to determine the number of solutions in positive integers of any number of linear equations with any number of variables. In applying this method I saw that the possibility of its success depended on the theorem named being true in a stricter sense than that used by its authors, of whom Euler verified but without proving the theorem by innumerable examples. As given by him, the theorem is this: every even number may be broken up in one or more ways into two primes.

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SYLVESTER, J. On the Goldbach-Euler Theorem regarding Prime Numbers. Nature 55, 196–197 (1896).

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