Abstract
<div class="line" id="line-21"> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with </span> <img src="http://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02737-8/images/img1.gif"/> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> prime factors for every </span> <img src="http://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02737-8/images/img2.gif"/> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes </span> <img src="http://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02737-8/images/img3.gif"/> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> with the property that </span> <img src="http://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02737-8/images/img4.gif"/> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> divides a highly composite </span> <img src="http://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02737-8/images/img5.gif"/> <span style='color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> . </span></div>
Original language | American English |
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Journal | Mathematics of Computation |
Volume | 83 |
State | Published - 2014 |
Keywords
- Carmichael numbers
- subset sum
Disciplines
- Theory and Algorithms
- Mathematics
- Number Theory