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Some problems on consecutive prime numbers

Published online by Cambridge University Press:  26 February 2010

P. Erdős
P. Erdős
Affiliation:
Department of Combinatories and Optimisation, University of Waterloo, Canada.
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Let 2 = p1 < p2 < … be the sequence of consecutive prime numbers. Put dn = pn+l − pn. Turán and I proved [1] that the inequalities dn+1 > dn and dn+1 < dn both have infinitely many solutions. It is not known if dn = dn+l has infinitely many solutions. The answer is undoubtedly affirmative but the proof will probably be very difficult [2]. It was a great surprise and disappointment to us that we could not prove that dn+2 > dn+1 > dn has infinitely many solutions. We could not even prove that (− 1)n (dn+l − dn) changes sign infinitely often. It seems certain that the answer to both of these questions is affirmative and perhaps a simple proof can be found.

Type
Research Article
Information
Mathematika , Volume 19 , Issue 1 , June 1972 , pp. 91 - 95
Copyright
Copyright © University College London 1972

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References

1.Erdős, P. and Turán, P., “On some new questions on the distribution of primes”, Bull. Amer. Math. Soc., 44 (1948), 271–278.Google Scholar
2.Erdős, P. and Turán, P., and Rényi, A., “Some problems and results on consecutive primes”, Simon Stevin (1949), 115–125.Google Scholar
3.Erdős, P. and Turán, P., “On the difference of consecutive primes”, Bull. Amer. Math. Soc., 44 (1948), 885–889.CrossRefGoogle Scholar
4.Erdős, P. and Turán, P., “Problems and results on the difference of consecutive primes”, Publicationes Delruen, 1 (1949), 37–39.Google Scholar