Succes in research, awful in solving in basic things I am a fourth-year Ph.D. student in pure mathematics. I am studying abroad. I need some advice.
Before getting through to my question, let me say something about myself. I am studying at a good university in Europe. I have 3 papers, which two of them are my own papers(without co-authors). I am planning to defend my thesis in the following summer. Moreover, I am applying for a postdoc position.

My Ph.D. story is a sad story. After two years, my Ph.D. supervisor informed the university that I do not know any things about math, and he could not work with me. Then, the university chose a co-supervisor for me. After 1year, I and my co-supervisor wrote a paper. To be honest, my co-supervisor had a lot of nice ideas and I wrote his idea in papers. I did not fill good because that was most of his idea, but my colleges told me we always did the same things. There was a lot of pressure on me because all of the people found out my problems with my supervisor and they thought I am weak and stupid. 

Last summer, I wrote two papers in the area which I have been working since my Ph.D. Then, everything changed. My supervisor said hello to me every day( before he did not say) and behavior everyone changes to me. I got some emails from great mathematicians about my papers. 

I have two big problems. First, I can not solve basic things in math. That makes me feel upset and worried. For instance, when I got up today, I was thinking the following statements are equal: For all x, there is y such that x+y=4&amp; There is x, for all y such that x+y=4, but somethings in my heart was saying I am wrong. So I start to read a logic book. That is not shaming for a Ph.D. student? Is it? or, Why I do not remember Linear Algebra? and other whys?

These kinds of questions make me feel I got lucky to write two papers. Anyone get lucky. My supervisor might be right, I do not know any things from math. How I can pass PhD exam?
I have seen my fair share of mathematicians failing basic tasks. Probably more iconic ones are related to arithmetics but that is more understandable.

For example I have seen a +40 years old working mathematician volunteering to teach differential equations because s/he didn't feel confident in his/her ability do solve them and wanted to force him/her-self to improve.

I don't think it is detrimental to fail basic tasks as long as you behave accordingly. As long as you discuss with others while doing your research and prepare before you teach, you can avoid most of the potential issues and fix them along the way.