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Comments - Friends, strangers, 7825 and computers
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[QUOTE="fresh_42, post: 5504334, member: 572553"] To start with: It's been a great pleasure to read your Insight. I wonder if Ramsey's theory will share a similar fortune as many other problems that could be formulated so easily. I'm always surprised again how some numbers or their properties fascinates us. When I think of Pythagoras I usually think of him as an ancient esoteric, a charlatan. On the other hand when it comes to Ramanujan I think of him as one of the greatest, possibly [I]the[/I] greatest virtuoso who ever played on the gamut of numbers. Does this discrepancy exist because Ramanujan contributed so much more to number theory than taxicab numbers or am I simply unfair to Pythagoras. It's also worth reading Simon Singh's book on[URL]https://www.amazon.com/dp/1841157910/?tag=pfamazon01-20[/URL]. It brought literally thousands of hobby mathematicians to try a solution because it could be stated so easily. Today it's believed that Fermat never had a real proof of it, or at best only for the case ##n=3.## One must know that at the time, some scholars made fun of it to write their colleagues letters, in which they stated to have proven something without actually given a proof, just to see whether the other one can find out. Nevertheless it was a real booster for number theory. The proof itself, however, is only understood by a handful of mathematicians (meanwhile maybe some more) and requires deep insights into the theory of elliptic curves and (as far as I know) the theory of modular forms. It is agreed to be proven, although only a few could verify this. What makes such a situation different from a computer based handling of cases? I remember I once had a computer proven some cases as well to verify a theorem. Long after it has been accepted, I found some loopholes in the program. It didn't change the result, for I could close them but nobody (except me) ever really noticed. Another prominent example is the Four-Color-Theorem. As in chess, I think we will have to develop a common ground on which we will judge computer based proofs. Personally I think it will require to say goodbye to some selfish human attitudes. Back to ##7825 = 5^2 \cdot P_{5\cdot P_{5+1}}##. What fascinates us on numbers? One can probably take any number and construct obscure relations around it. I like the hypothesis that it comes from our evolutionary based need to recognize patterns. It is kind of fascinating by itself that this pattern-recognition led to so many theorems and knowledge in number theory. And that it has such an enormous history from Pythagoras, over Ramanujan to Wiles. [/QUOTE]
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