A Proof of the Celebrated Goldbach’s Theorem Doron ZEILBERGER^{1}

Theorem (Goldbach [3], see also [1][2][4])

∑_{m,n≥2}′ \frac{1}{m^{n} − 1} = 1 ,

where ∑′ means that every term only occurs once (for example 1/15 = 1/(16 − 1) is only added once even though 16 = 4^{2} = 2^{4}). Proof: Let R denote the set of all integers larger than 1 that are not perfect powers: R = {2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, …}. Since every perfect power can be written uniquely as r^{s} (r ∈ R, s ≥ 2) and every integer ≥ 2 can be written uniquely as r^{s} (r ∈ R, s ≥ 1), we have

∑_{m,n≥2}′ \frac{1}{m^{n} − 1} = ∑_{r∈R} ∑_{s=2}^{∞} \frac{1}{r^{s} − 1} = ∑_{r∈R} ∑_{s=2}^{∞} ∑_{i=1}^{∞} \frac{1}{r^{si}} = ∑_{r∈R} ∑_{i=1}^{∞} ∑_{s=2}^{∞}(\frac{1}{r^{i}})^{s} =

∑_{r∈R} ∑_{i=1}^{∞} \frac{(\frac{1}{r^{i}})^{2}}{1 − \frac{1}{r^{i}}} = ∑_{r∈R} ∑_{i=1}^{∞} \frac{1}{r^{i}(r^{i} − 1)} = ∑_{m=2}^{∞} \frac{1}{m(m − 1)} = ∑_{m=2}^{∞} (\frac{1}{m − 1} − \frac{1}{m}) = 1 .

References

1. E. Catalan, Note sur la sommation de quelques séries, Journal de Mathématiques Pures et Appliquées 7 (1842), 1-12.

2. G. Chrystal,“Algebra ”, Part II, reprinted by Chelsea, N.Y. 1964, [p. 422].

3. C. Goldbach, Letter to L. Euler, 1737.

4. R. L. Graham, O. Patashnik and D.E. Knuth, “Concrete Mathematics”, Addison Wesley, 1989, [p. 66].

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg . April 1, 2006.