20180102, 03:09  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{6}·31 Posts 
PseudoPrime Bases
Hi all,
Is there a systematic method (other than brute force) to find a base whereby a given composite n will yield a false positive in FLT testing, other than n+1 and n1 ? Does such a base exist for any given composite? Are such Bases finite in number? Thank you in advance. Last fiddled with by a1call on 20180102 at 03:11 
20180102, 03:52  #2 
Jun 2003
2^{3}×607 Posts 

20180102, 04:15  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
11111000000_{2} Posts 
Thank you for the insight axn.
Re: My 2nd question, well it certainly can't be infinite unless bases greater than n are considered. So a mute question I can see. 
20180102, 06:03  #4  
Aug 2006
2^{2}×1,493 Posts 
Does "FLT testing" mean the Fermat probableprime test? I'll assume this below, otherwise please clarify the term.
Quote:
If \[n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}\] (with p1 < p2 < ... < pk and all exponents positive integers) then there are \[\prod_{i=1}^k \gcd(p_i1, n1)\] bases to which n is a pseudoprime. This fits axn's comment pretty well: it's easy to see that only prime powers could have the minimum 2 bases, and p  1 divides p^e  1 so only powers of 3 work. As you pointed out, this question is moot  there are only finitely many choices unless you count integers rather than integers mod n. 

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