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Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
I plotted these for the first fifty functions or so once. If I recall correctly, for high k, they seem to trace out a vaguely ellipse-shaped curve in the complex plane. At the time, I didn't know how to interpret this, and I suppose I still don't. It would be interesting to try to work out a limiting distribution of the roots (you might need to rescale, or it might not exist), and think about what that distribution says about these sums, Bernoulli numbers, and so on.tomtom2357 wrote:What are the roots of the functions?
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
tomtom2357 wrote:At what point do the roots become complex?
Proginoskes wrote:(I assume that other theorem are allowed here.)
How about Fürstenberg's proof of the infinitude of primes?
http://en.wikipedia.org/wiki/F%C3%BCrst ... _of_primes
(or http://www.cut-the-knot.org/proofs/Furstenberg.shtml , since Wikipedia is going away in a day or so.)
antonfire wrote:I plotted these for the first fifty functions or so once. If I recall correctly, for high k, they seem to trace out a vaguely ellipse-shaped curve in the complex plane. At the time, I didn't know how to interpret this, and I suppose I still don't. It would be interesting to try to work out a limiting distribution of the roots (you might need to rescale, or it might not exist), and think about what that distribution says about these sums, Bernoulli numbers, and so on.tomtom2357 wrote:What are the roots of the functions?
Blatm wrote:In an upcoming assignment I have to prove that the sum of consecutive squares is n(n+1)(2n+1)/6.
skeptical scientist wrote:Here's another proof that there are infinitely many primes, using ideas of compression. The basic idea is if there were only n primes, we could use prime factorization to compress every k-bit string into n log(k+1) bits, violating the pigeonhole principle.Spoiler:
vadsomhelst wrote:[...]
Xanthir wrote:skeptical scientist wrote:Here's another proof that there are infinitely many primes, using ideas of compression. The basic idea is if there were only n primes, we could use prime factorization to compress every k-bit string into n log(k+1) bits, violating the pigeonhole principle.Spoiler:
That one is really awesome. I love it!
phlip wrote:I remember one I read in another thread on here a while back: A proof that the nth-root of 2 is irrational for n>2:
Say there was a p/q, p and q both integers, such that (p/q)^{n} = 2. Then:
p^{n}/q^{n} = 2
p^{n} = 2q^{n}
q^{n} + q^{n} = p^{n}
However, this violates Fermat's Last Theorem, and thus has no solutions.
The proof of the statement for n=2 is left as an exercise for the reader.
tomtom2357 wrote:This is possibly a circular proof, because fermat's last theorem could rely on this
Hopper wrote:tomtom2357 wrote:This is possibly a circular proof, because fermat's last theorem could rely on this
It almost certainly does not. Though I'm pretty sure it uses the fact that the integers are integrally closed, which immediately implies nth roots of 2 are irrational.
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