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On the order of magnitude of Jacobsthal's function

Published online by Cambridge University Press:  20 January 2009

R. C. Vaughan
R. C. Vaughan
Affiliation:
Imperial College, London, S.W.7.
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Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, if

then

It is probably true that

where ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 4 , September 1977 , pp. 329 - 331
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

REFERENCES

(1) Erdös, P., On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal, Math. Scand. 10 (1962), 163–170.CrossRefGoogle Scholar
(2) Iwaniec, H., On the error term in the linear sieve, Ada Arithmetic 19 (1971), 1–30.CrossRefGoogle Scholar
(3) Jacobsthal, E., Über Sequenzen ganzer Zahlen von denen keine zu n teilerfremd ist, I-III, Norske Videnske. Selsk, Forh. Trondheim 33 (1960), 117–139, IV,Google Scholar
Jacobsthal, E., Über Sequenzen ganzer Zahlen von denen keine zu n teilerfremd ist, I-III, Norske Videnske. Selsk, Forh. Trondheim 34 (1961), 1–7, V,Google Scholar
Jacobsthal, E., Über Sequenzen ganzer Zahlen von denen keine zu n teilerfremd ist, I-III, Norske Videnske. Selsk, Forh. Trondheim 34 (1961), 110–115.Google Scholar
(6) Kanold, H.-J., Über Primzahlen in arithmetischen Folgen, Math. Ann. 156 (1964), 393–395, II,CrossRefGoogle Scholar
Kanold, H.-J., Über Primzahlen in arithmetischen Folgen, Math. Ann. 156 (1964), 393–395, II, 157 (1965), 358–362.CrossRefGoogle Scholar
(8) Kanold, H.-J., Über eine zahlentheoretische Funktion von Jacobsthal, Math. Ann. 170 (1967), 314–326.CrossRefGoogle Scholar