Why is 0x10001 used as an RSA exponent so often?
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.