filifa
/
eulerbooks
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases 1 Wiki Activity

change link to avoid stray parenthesis

This commit is contained in:
filifa 2025-07-19 00:59:54 -04:00
parent 3f42328cb3
commit a2dcf2aaeb
1 changed files with 1 additions and 1 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
notebooks/problem0007.ipynb
View File

@ -48,7 +48,7 @@
"\n",
"Consider the fact that $3^{6} \\equiv 3 \\pmod{6}$, but 6 is composite. In cases like this, we say that $p$ is a *base $a$ Fermat pseudoprime*, e.g. 6 is a base 3 Fermat pseudoprime. 6 is also a base 4 Fermat pseudoprime, but 2 and 5 both serve as witnesses to 6 being composite (we need only consider bases modulo $p$).\n",
"\n",
"This may lead us to believe that we can just check every base $a$, and if there are no witnesses, then $p$ is prime. However, (along with this method being inefficient for large $p$), there are numbers that are Fermat pseudoprimes for *every* base, called [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number). Therefore, the [converse](https://en.wikipedia.org/wiki/Converse_(logic)) of Fermat's little theorem does not hold.\n",
"This may lead us to believe that we can just check every base $a$, and if there are no witnesses, then $p$ is prime. However, (along with this method being inefficient for large $p$), there are numbers that are Fermat pseudoprimes for *every* base, called [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number). Therefore, the [converse](https://w.wiki/BAyj) of Fermat's little theorem does not hold.\n",
"\n",
"Nevertheless, Carmichael numbers are relatively rare, so if you pick a random number to test and a few random bases, and find no witnesses, you can be pretty sure the number is in fact prime.\n",
"\n",