20220124, 18:16  #12 
Feb 2017
Nowhere
5770_{10} Posts 
Let us not forget the numbers that have changed status.
Fermat (letters, 1630's  1640's) claimed that all numbers 2^(2^{n}) + 1 were prime. Thus, 2^{32} + 1 changed from prime to composite (Euler, 1732; published 1738) Mersenne claimed 2^{p}  1 was prime (at least for p ≤ 257) when p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. Thus, 2^{61} 1 changed from composite to prime (Peruvshin, 1883) 2^{67}  1 changed from prime to composite (Cole, 1903) etc 
20220124, 19:54  #13 
Jun 2015
Vallejo, CA/.
10001000101_{2} Posts 
In this line of thought –and this may has been asked and answered before– is there any pseudoprime to many bases (but not a Carmichael number) that was assumed to be a PRP and was later confirmed as composite?

20220124, 19:55  #14 
Jun 2021
2F_{16} Posts 
prime numbers boring. take a look into prime angles

20220124, 21:07  #15  
Feb 2017
Nowhere
2×5×577 Posts 
Quote:
A PRP is generally understood to mean any integer greater than 1 that does not test as composite to some base(s) with a Fermat (or Euler or MillerRabin) test, while a "pseudoprime" is generally understood to mean a composite number that does not so test as composite. I don't know about "assumed to be prime because they tested as PRP's but then proved composite." I don't know of any examples offhand. There might be "mundane" examples of numbers that individual users of some software package assumed to be prime because it tested as a PRP with the package's test, but later proved to be composite. I note that though a Carmichael number n is guaranteed to test PRP by a Fermat test to any base prime to n, it may well be revealed as composite by a MillerRabin test. In such a case, MillerRabin provides a square root of 1 other than 1 or 1, hence a proper factorization. For example, Mod(2,561)^560 and Mod(2,561)^280 are both Mod(1,561), but Mod(2,560)^140 = Mod(67,561). Then gcd(671,561) = 33 and gcd(67+1,561) 17 are proper factors. There are, however, ways of constructing numbers which are the product of two prime numbers, that will test as PRP's by MillerRabin to any given finite set of (usually prime) bases. But who knows? The next M_{p} that tests PRP to base 3 may be proven composite by an LL test. 

20220124, 21:28  #16 
Sep 2002
Database er0rr
1000000011010_{2} Posts 
https://primes.utm.edu/notes/proofs/a_pseudoprimes.html shows how to construct basea pseudoprimes.
https://www.fq.math.ca/Scanned/414/chen.pdf and https://www.d.umn.edu/~jgreene/baillie/BailliePSW.html supposedly shows how to construct BPSW pseudoprimes http://pseudoprime.com/dopo.pdf Pomerance argues for the infintude of BPSW pseudoprimes. More at: https://en.wikipedia.org/wiki/Bailli...est#References Last fiddled with by paulunderwood on 20220124 at 21:29 
20220124, 21:33  #17  
Jun 2015
Vallejo, CA/.
2105_{8} Posts 
Quote:
Yes, I messed up when using the word pseudoprime. What I meant was along he line of a number that after many tests and varied approaches is indistinguishable from a true PRP but is eventually found to be composite. I.e pseudoprime. I had read somewhere of this but I can’t remember where. 

20220130, 04:48  #18  
Romulan Interpreter
"name field"
Jun 2011
Thailand
7·1,423 Posts 
Quote:
I mean, if so many big guys made mistakes... 

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