Number Theory Update

I haven't found a solution to 13^n (mod n) = 36 yet, but not without a good bit of searching. I've run the factorization method using ECM, and variants of the prime and line algorithm, including a two-prime version, all without success.

Sometimes that's the case with these equations though, like with 2^n (mod n) = 465. I think that took me about a week with focus on it to find n=164196324252985941533.

It's another open problem whether solutions exist to it. I'm pretty sure they do, but the reason I think some are much harder than others is because of the primes that cannot divide n for specific equations.

In 2^n (mod n) = 465, you can show that if p divides n, then p cannot be 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43. Only when you get to 47 is that a possible prime factor. Because of all these small primes being impossible, you have a huge range of n eliminated, and cause other primes to get eliminated -- when GCD(order2p, log2 of 465 mod p) contains a bad factor.

For 13^n (mod n) = 36, only the primes 23, 53, 79 below 107 are possible divisors of n. So it's not surprising a small solution does not exist (Max Alekseyev reported n must be > 10^17).