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Mathematics 2014, 2(1), 37-52; doi:10.3390/math2010037

Bounded Gaps between Products of Special Primes

1,*  and 2,*
1 Massachusetts Institute of Technology, 305 Memorial Drive, Cambridge, MA 02139, USA 2 University of California, Berkeley, 1676 S. Blaney Ave. San Jose, CA 95129, USA
* Authors to whom correspondence should be addressed.
Received: 23 August 2013 / Revised: 18 February 2014 / Accepted: 25 February 2014 / Published: 3 March 2014
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In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case.
Keywords: bounded prime gaps; square-free numbers; modular elliptic curves bounded prime gaps; square-free numbers; modular elliptic curves
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Chung, P.N.; Li, S. Bounded Gaps between Products of Special Primes. Mathematics 2014, 2, 37-52.

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