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I sometimes wondered why mathematicians are so pedantic about proofs. Why can't we just check the first n cases with a supercomputer and do without rigorous proofs? Here's an excellent example of why proofs matter.
Consider this proposition: 313(x3+y3)=z3 has no solution when x,y,z∈Z+.
If you check with a computer, then this proposition will appear true.
The smallest counterexample has more than 1000 digits, and your computer program might not have checked that far. If you relied on the assumption that this equation has no solution in a cryptographic application without a proof, the consequences could be catastrophic.
That's fascinating. Thanks for sharing. Number theory is mindblowing. The integers look very simple at first glance, but they contain immense complexity and structure.