Why is 0x10001 used as an RSA exponent so often? Are there any other commonly used exponents? Why are they selected?
Here is the beginning of that answer from info security stackexchange for the reader’s convenience:


  There is no known weakness for any short or long public exponent for RSA, as long as the public exponent is "correct" (i.e. relatively prime to p-1 for all primes p which divide the modulus).
  
  If you use a small exponent and you do not use any padding for encryption and you encrypt the exact same message with several distinct public keys, then your message is at risk: if $e = 3,$ and you encrypt message $m$ with public keys $n_1,n_2,n_3$ then you have $c_i= m^3 \pmod{n_i}$ and use the chinese remainder theorem to recover the message by a
  non modular cube root extraction.
  
  The weakness, here, is not the small exponent; rather, it is the use of an improper padding (namely, no padding at all) for encryption.