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Richert’s theorem

In 1949 H.-E.Richert proved  [1]  that every positive integer typeset structure is a sum of distinct primes. The proof was based on Bertrand’s postulate in the form that for typeset structure there is always a prime typeset structure such that typeset structure. In   [2]  he extended the proof to get the same conclusion for squarefree, triangular and pentagonal numbers.

W.Sierpinski  [3]  p. 152 further developed Richert’s argument to more general sequences:

Theorem. Let  typeset structure be a sequence of positive integers such that

(1) there exists positive integer typeset structure with  typeset structure for typeset structure,

(2) there exist positive integers typeset structure and typeset structure such that each of the numbers typeset structure is the sum of distinct terms of the sequence typeset structure.

Then typeset structure is complete.

Related results can be found in  [4] .

References

[1]  Richert, H. E. (1949). Über Zerfällungen in ungleiche Primzahlen. (German). Math. Z., 52, 342-343.

[2]  Richert, H. E. (1949). Über Zerlegungen in paarweise verschiedene Zahlen. (German). Norsk Mat. Tidsskr., 31, 120-122.

[3]  Sierpinski, W. (1988). Elementary theory of numbers. Transl. from the Polish. Edited by A. Schinzel. 2. ed.. Amsterdam - New York - Oxford: North-Holland; Warszawa: North-Holland Mathematical Library, Vol. 31; PWN - Polish Scientific Publishers.

[4]  Brown, jun., J. L. (1976). Generalization of Richert's theorem. Am. Math. Mon., 83, 631-634.

Cite this web-page as:

Štefan Porubský: Richert’s theorem/i>.

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