I have placed a bet that I can create a public key such that my adversary will not be able to crack (decrypt) it for at least one week. For my primes $p$ and $q$, I chose very large numbers that are $101$ digits long, each. I also tried to make them to be as distinct as possible.
Now what I am wondering is whether to choose a big encryption exponent or a small one. Since I have began my study of Cryptography about a month ago, I am not too familiar with choosing a solid encryption exponent. I heard that choosing the digits $3, 5, 17, 257$ or $65537$ would give me a good security but I have not read about it. I was thinking of going with $257$ because intuition tells me that there exists a inverse relation between the two: when choosing large primes for $p$ and $q$, then pick a small encryption exponent and vice versa. But I don't know if this makes complete sense in some regards.
Any input is appreciated.