a^x=x (mod p)
 

Are there any results concerning the following problem: given p - prime, 1<a<p find 0=<x<p such that a^x = x (mod p)?

There are 6 before you even enter double digits.[code]2^3 = 3 mod 5
3^2 = 2 mod 7
3^4 = 4 mod 7
3^5 = 5 mod 7
4^2 = 2 mod 7
4^4 = 4 mod 7[/code]Generated in [url=http://pari.math.u-bordeaux.fr/]PARI/GP[/url] with:[code]c=0;

forprime(p=2,10, for(a=2,p, for(x=0,p, if(lift(Mod(a^x,p))==x, c++;print(a,"^",x," = ",x," mod ",p) ) )));

print(c);[/code]There are 38 results for p<20, 969 for p<100 and 75434 for p<1000, but I won't list them here since you have the code now.

Edit: Limiting a and x to primes aswell produces 3 results with p<10, 12 with p<20, 101 with p<100 and 2750 with p<1000.

Edit2: The highest result for p<1000 is 995^803 = 803 mod 997.