Please let me know if I wasn't allowed to answer my own question, in particular, since the answer is essencially somebody else's idea. I thought the body of the post is too long at the moment, and it won't be a good idea to make it longer. Moreover, I am about to report to our next meeting what I have learnt so far. Again, I thought it is also a good idea to let the viewers see what I saw so far in a neutral way. I have some non-neutral comments that I will write them in meta as soon as they find a reportable form.

Who knows, perhaps some of the younger viewers find a chance to check Zeilberger's claim in 2100 :)

Second quote:I can envision an abstract of a paper, c. 2100, that reads : “We show, in a certain precise sense, that the Goldbach conjecture is true with probability larger than 0.99999, and that its complete truth could be determined with a budget of $10B.”...As absolute truth becomes more and more expensive, we would sooner or later come to grips with the fact that few non-trivial results could be known with old-fashioned certainty. Most likely we will wind up abandoning the task of keeping track of price altogether, and complete the metamorphosis to non-rigorous mathematics.

First quote: There are writings on the wall that, now that the silicon savior has arrived, a new testament is going to be written. Although there will always be a small group of “rigorous” old-style mathe- maticians(e.g. [JQ]) who will insist that the true religion is theirs, and that the computer is a false Messiah, they may be viewed by future mainstream mathematicians as a fringe sect of harmless eccentrics, like mathematical physicists are viewed by regular physicists today. [JQ: Jaffe and Quinn]

Here are two quotes from "Theorems for a price". I have chosen the first quote for the language used, that is highly related to the form of the question posed. The second quote comes to give a glimpse of Zeilberger's "answer" to the question (i.e, assuming he was answering the question). Thanks to "unknown" and John. Both quotes have been just chosen to give a quick view of the paper to the viewers.

who are interested in "mathematics education". It seems that we live in two different cultures. The best thing I can say about the other culture is people there try to experience education by themselves. It is not a bad thing until you read something written for example in Monthly discovering a point very well-known for other culture. It may have many different reasons, for example, it is a well-known problem of mathematics education that people do not use the same term for more or less the same idea. In sum, I believe the two cultures should find a way to communicate better.

@Pete L. Clark, Dear Pete, thanks for your advice. But, I am afraid to say I don't think that works whether they accept "questions" or not. It is hard to live in one culture trying to understand another culture. Let me explain. I am a mathematics educator with a sound background in mathematics. When within mathematics education community we have no problem to talk about mathematics and what mathematicians do. We understand each other, to say the least. On the other hand, I work in a mathematics department, having many good friends in mathematics community (to be continued!)