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York concluded that: "Both sides in the arms race are confronted by the dilemma of steadily increasing military power and steadily decreasing national security.
An implicit and almost universal assumption of discussions published in professional and semipopular scientific journals is that the problem under discussion has a technical solution.
For nearly 100 years, the conjecture had confused the sharpest minds in maths, many of whom had claimed its proof, only to have their work discarded upon subsequent scrutiny. By the time that Perelman defeated the conjecture, after many years of concentrated exertion, the Poincaré had affected him so profoundly that he appeared broken too.
He had shut off contact with most friends and colleagues, stopped cutting his hair and nails and cultivated a wild beard.
Because of previous failures in prophecy, it takes courage to assert that a desired technical solution is not possible.
Wiesner and York exhibited this courage; publishing in a science journal, they insisted that the solution to the problem was not to be found in the natural sciences.
If you don't remember the Pythogrean theorem from high school geometry, it's the one that says that for any right triangle, a Got it?
It turns out that there are solutions for the question up to 7,824, but as soon as you try to do it for 7,825, you fail. Proving it took a dataset of 200 TB, blowing away the previous largest proof of 13 gigabytes.They cautiously qualified their statement with the phrase, "It is our considered professional judgment...." Whether they were right or not is not the concern of the present article.Rather, the concern here is with the important concept of a class of human problems which can be called "no technical solution problems," and more specifically, with the identification and discussion of one of these.It's like there's an election coming and you have to pick one side or the other, and stay that way. Carry this logic forward into much bigger numbers and you could see where this would start to get tricky.The boolean Pythagorean triples problem, as put forth by mathematician Ronald Graham in the 1980s, asks whether, in this two-color scenario, you could color the numbers so that no set of Pythagorean triples are all the same color—that is, all three red or all three blue. What makes it so hard is that one integer can be part of multiple Pythagorean triples. If 12 has to be red in that 5-12-13 triple, it might force changes down the line that would result in a monochrome triple somewhere. The answer to Graham's question—could one color the numbers so that no set of Pythagorean triples are all the same color—is no, but proving it requires a computer to work out via brute force calculations all of those combinations of numbers and colors.