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Additive Prime Number Theory*

Published online by Cambridge University Press:  03 November 2016

L. Mirsky
L. Mirsky*
Affiliation:
The University of Sheffield
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Extract

The scope of the additive prime number theory is evident from its name. In this part of the theory of numbers we are concerned with the representation of integers as sums of primes; and here the central problem consists in the proof (or possibly refutation) of a celebrated conjecture made by Goldbach in 1742 to the effect that every even integer ≥4 can be represented as the sum of 2 primes. The truth of this conjecture is still unsettled; but though progress in this field has been slow, attempts to deal with the problem have led to a whole series of striking results.

Type
Research Article
Information
The Mathematical Gazette , Volume 42 , Issue 339 , February 1958 , pp. 7 - 10
Copyright
Copyright © Mathematical Association 1958

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Footnotes

*

Shortened version of an address given to the British Mathematical Colloquium at St. Andrews in September, 1956.

References

Page 7 of note † This hypothesis, which we shall refer to as H, is a generalization of the famous “Riemann hypothesis.”

Page 7 of note ‡ The symbol ~ in (1) indicates that the ratio of the two sides tends to unity. The letter p is reserved for primes throughout; and in the fist product on the right-hand side of (1) p ranges over all primes, while in the second it ranges over all prime divisors of n.

Page 8 of note * The letter c is reserved for absolute positive constants.

Page 9 of note * The density of the sequence {s i} is defined as the lower bound of S(n)/n, where S(n) denotes the number of s i which do not exceed n.

Page 9 of note † A. Selberg stated in 1949 that it is possible to replace the number 4 in Theorem 4 by 3, but he has not published a detailed proof.