I am writing a (high-school) paper on the public key exponent's (in textbook RSA - no padding is discussed!) significance in terms of time and security. The time part is done; as for the security part, so far I have shown example of various attacks that can take place when the public key is low (e.g. $e$ root attack, Håstad Broadcast Attacks etc).
My question is: Is there a way to mathematically justify that there is no way to prove that the security of the system lies in the value of the public key exponent?
I have also done some readings on the integer factorization problem whose intractability cannot be proven - can I relate this to the link between security and the public key?