Abstract
AbstractWe show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput.29 (1975), 79–81] and Sun [‘On integers not of the form ±pa±qb’, Proc. Amer. Math. Soc.128 (2000), 997–1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results.
Publisher
Cambridge University Press (CUP)
Reference23 articles.
1. On the largest prime factor of a Mersenne
number
2. [19] Sloane N. J. A. , ‘The On-Line Encyclopedia of Integer Sequences’, published electronically atwww.research.att.com/∼njas/sequences/, 2007.
3. Composites that remain composite after changing a digit;Fileseta;J. Comb. Number Theory,2010
4. Problems
5. Fermat numbers and integers of the form ak+al+pα
Cited by
24 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献