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On the easier Waring problem for powers of primes. II

Published online by Cambridge University Press:  24 October 2008

P. Erdös
P. Erdös
Affiliation:
The UniversityManchester
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Extract

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:

The density of each of the sets of integers

is positive.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 35 , Issue 2 , April 1939 , pp. 149 - 165
Copyright
Copyright © Cambridge Philosophical Society 1939

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References

† See, for example Landau, E., Göttinger Nachr. (1930), 255–76,Google Scholar Satz 21.

‡ Vinogradov has recently proved that every large number of the form 24k + 5 is the sum of 5 squares of primes. See C.R. Acad. Sci. U.R.S.S. 16 (1937), 131–2.Google Scholar

* Titchmarsh, E. C., Rendiconli Circ. mat. Palermo, 54 (1930), 416.Google Scholar

* Ramanujan, S., Collected papers (Cambridge, 1927), pp. 262–75.Google Scholar

* S. Ramanujan, op. cit.

* a′i and a′j here run through the integers of Lemma 6.

† π(x) denotes the number of primes not exceeding x.

* Erdös, P., Quart. J. Math. 6 (1935), 205–13.CrossRefGoogle Scholar

* S. Ramanujan, op. cit.

† J. London Math. Soc. 10 (1935), 126–8.Google Scholar

‡ J. London Math. Soc. 11 (1936), 92–8.Google Scholar

* For it follows from Lemma 3 of I that the number of solutions of α1 = p′ν − p ν with is less than