What is the most recommended quality a mathematician should have? Extremely high IQ levels?Passion for what he does?Patience and "stuborness"?Something else? Of course all of them are necessary ,but which of them do you believe is the most needed ?
People asked Gauss: "how come you always see things that we don't?" And Gauss replied: "If only you would have thought about it for as long as I have..."
On another occasion someone asked Gauss about Fermat's last theorem. Gauss said he didn't care much, since he could easily fabricate a number of similar conjectures that one could neither proof nor disproof.
I think that's the most vital aspect: tackle problems you think are just on the boundary of your reach and never ever ever give up. (Unless you're getting danderously close to insanity. But that's another story.)
A mathematician is interested in the Concepts, the "principles" behind things. It is important (IMO) to focus on the big ideas, on Understanding, as opposed to procedures and memorization.
I don't think a high IQ is especially important but most mathematicians I know enjoy playing with ideas (any ideas!) and so enjoy intellectual games or intellectual exercises.
Most of us are (eventually) quite passionate about mathematics -- but that passion probably comes later?
And many of us hate memorizing -- I got into math partly because I could do it without any memorization. If I understood the concepts, it all made sense and no memorization is necessary.
First of all, like what you do. I'd even say love what you do.
Second, be monomaniacal. Just to tell a little anecdote, a cousin of Albert Einstein once sent him a letter asking why her husband, a mathematician, was not making great discoveries like Albert. Albert replied that was because her husband was not monomaniacal about mathematics. This story tells us that if you want to attain the summit of your profession, you should also put all your efforts into it.
Third, nowadays, it is very rare for mathematicians, and scientists in general, to work in isolation. Most papers that are published have multiple authors, collaboration is getting more and more important. Definitely communication skills and the ability to collaborate have become paramount.
Fourth, you'll also have to learn how to sell your product. This is sadly a result of how research is becoming increasingly funded. You'll be up against a whole lot of bureaucracy.