Proof: Otherwise you could break X

⬛ Some of you may be familiar with this situation. From time to time you find yourself thinking about a new approach to a problem that you know is difficult. And although you know that you should
probably be doing something more productive, you try new ideas to tackle the problems.

And every now and then you stumble across a more or less basic fact that prevents your idea from working. I will try to give some examples that may surprise you. I will focus on the three main problems: discrete logarithm problem, factorization problem and the class number computation problem.

Often the examples come from situations that similar to:

    Alice: "I have a great idea. What if you could find some object O that has property P and Q."

    Bob: "Perfect, O is well known and P and Q seem not that restrictive."

A few days later.

    Alice: "I can not find such an object. All objects either have P or Q but not both."

    Bob: "Yeah, you are right, probably there is some theorem that we don't know."

There are probably many more examples or better ones. But at least all of the three examples below crossed my road during the last years.