TITLE: Is there a nice proof that $123456789098765432111$ is prime?
QUESTION [45 upvotes]: The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact 

$$12345678987654321 = 111111111 \times 111111111$$ 

which is of course well-known.) 
By "proof", I assume he means something more humanly illuminating than asking a computer program. I haven't a clue what he has in mind. Does someone have an idea?

REPLY [3 votes]: More Fun with Numbers
The number 12345678987654321=111111111x111111111 is mentioned in the OP. And, believe it or not,  $1234567898765432111$ is a prime number too!!!
You have to believe it! The following $maxima$ output shall demonstrate this using Pratt's primality certificat.

(%i1) m:1234567898765432111;a:29;power_mod(a,m-1,m);factor(m-1); makelist(power_mod(a,(m-1)/p,m),p,map(first,ifactors(m-1)));
(%o1) 1234567898765432111
(%o2) 29
(%o3) 1
(%o4) 2*5*11*11223344534231201
(%o5) [1234567898765432110, 972745681039223016, 1223153431857342200, 1089307852892054661]


%i1 is the input line. 
%o1is the output showing the modul $m$, the number that we check if it is prime. 
%o2 is an outputline showing the number $a$ we exponate. 
%o3 is the output of $a^{m-1} \pmod{m}$. It should be $1$ if $m$ is a prime. But it could be $1$ even if $M$ is composite.
%o4 is the factorisation of $m-1$ and
%o5 are the numbers $a^{\frac{m-1}{p}}$ for all prime factors of $m-1$

The factors $2$,$5$ and $11$ are primes but it is not obvious that $11223344534231201$ is a prime number. so we add a proof that $11223344534231201$ is a prime too:

(%i1) m:11223344534231201;a:3;power_mod(a,m-1,m);factor(m-1); makelist(power_mod(a,(m-1)/p,m),p,map(first,ifactors(m-1)));
(%o1) 11223344534231201
(%o2) 3
(%o3) 1
(%o4) 2^5*5^2*7*73*101*109^2*137*167
(%o5) [11223344534231200, 7251767246727978,1687389182360412, 5679768249246961, 2181793601204580, 9078160829860754, 8735272021960592, 1320423471360269]

Because it is easy to check that $2,5,7,73,109,137,167$ are primes we are finished.
( both $a=29$ and $a=3$ can be replaced by the larger but funnier number $1111111$ int the certificates)
But how does this primality certificate work?
The following is well known:

all residue classes $a$ that are relatively prime to the module $m$ constitute a multiplicative  group $\pmod{m}$  
If $a^{m-1} \pmod{m}$ is not equal to $1$ then $m$ could not be a prime because this contradicts Fermat's little theorem.  
$\text{ord}_{m-1}(a)$  is the smallest power $e$ such that $a^{e} \equiv 1 \pmod{m}$, it  is a divisor of every such $e$. Especially of $\phi(m)$.  
if $a^{m-1} \equiv 1 \pmod{m}$ then  $\text{ord}_{m-1}(a)$ is a divisor of $m-1$. If it is a proper divisor of $m-1$ it must be a divisor of $\frac{m-1}{p}$ for a prime $p$ dividing $m$. Then $a^\frac{m-1}{p} \equiv 1 \pmod{m}$ for such a $p$   
all the residue classes $a,a^2,...,a^{\text{ord}(a)}$ are pairwise different

So if $\text{ord}(a)=m-1$ and $\phi(m)=m-1$ and therefore $m$ is a prime.
@ronno would note this certificate  as
1234567898765432111,29,1234567898765432111-1=2*5*11*11223344534231201
  11223344534231201,3,11223344534231201-1=2^5*5^2*7*73*101*109^2*137*167
    167,...
    137,...
    109,...
    101,...