
Answer to a 150-Year-Old Math Conundrum Brings More Mystery - prostoalex
http://www.wired.com/2015/06/answer-150-year-old-math-conundrum-brings-mystery/
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mturmon
Sometimes my instinct is to scoff at the practical value of asymptotic
existence results like this, but there's a quote in the article that provides
a good counterpoint:

Keevash’s result will shift the mindset of mathematicians who are trying to
find designs, Colbourn said. “Before, it wasn’t clear whether the focus should
be on constructing designs or proving they don’t exist,” he said. “Now at
least we know the effort should focus on constructing them.”

I only glanced at the paper, but perhaps this is analogous to Shannon's
original proof of existence of codes meeting channel capacity -- which was
also a non-constructive, asymptotic result.

That pointed the way for people to start working out channel codes, and after
_only_ a few decades of work (say, 1948 to 1993), turbo codes were discovered.

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furyofantares
When doing math proof exercises as an undergrad I found problems of the form
"Prove X" or "Disprove X" to be much easier than "Prove or disprove X" despite
no noticeable difference in the difficulty of understanding the proof once
constructed. I felt that prove/disprove was significantly more than twice as
hard, actually. I thought I could trick myself into making it between 1 and 2
times as hard by attempting to force myself to believe the problem was one
form or the other for set periods of time, and that did seem to help some, but
not nearly to the extent I had hoped.

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reagency
At undergrad, a successful technique is to assume X exists and try to
construct it. As you do so, you either find one or reach a contradiction to
every possible class of structure.

Nonconstructive proofs of existence in rare, outside of logic classes. Most
branches of undergrad math aren't interested in mere existence

~~~
thaumasiotes
The famous proof that the primes are infinite is a nonconstructive existence
proof. And weirdly, it proceeds entirely by trying to construct a new prime.
It just fails to do so (well, sort of) while still invalidating the idea of an
upper bound.

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Dylan16807
I was going to say it's constructive but apparently the wording matters.

this version is nonconstructive: assume you have all primes -> multiply, add
one -> contradiction

this version is constructive: you have n primes -> multiply plus one is always
a new prime

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panic
This constructive version doesn't work, though. The "new prime" 2×3×5×7×11×13
+ 1 actually equals 59×509.

~~~
makomk
All of its prime factors are in fact new primes though - the constructive
proof just requires you to use the prime factors of all the primes multiplied
together, plus one.

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schoen
This reads something like a classic Martin Gardner column -- clear,
tantalizing, and presenting a highly technical math topic helpfully for non-
specialists. Way to go, Erica Klarreich.

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cromwellian
You can see an old puzzle I concocted based on block designs and projective
planes here:
[http://cromwellian.blogspot.com/search/label/puzzles](http://cromwellian.blogspot.com/search/label/puzzles)

Solution at top, so scroll to the bottom first.

There are practical applications. :)

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thret
While it is trivial to find a single solution to the problem presented at the
end of this article, does anyone have a nice general one? Please share if so.

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selimthegrim
IIRC Keevash used to teach at Caltech and was Rick Wilson's colleague.

