ITT: Cool elementary paradoxes/unintuitive stuff
>>7729491
In general rings you can have much smaller examples. Zero char fields a shit
>>7729491
>>7729491
[math]\int_0^{\infty} \mathrm{sinc}\left(\frac x1\right) dx=\frac{\pi}2[/math]
[math]\int_0^{\infty} \mathrm{sinc}\left(\frac x1\right)\cdot \mathrm{sinc} \left(\frac x3\right)dx=\frac{\pi}2[/math]
[math]\int_0^{\infty} \mathrm{sinc}\left(\frac x1\right)\cdot \mathrm{sinc} \left(\frac x3\right)\cdot \mathrm{sinc}\left(\frac x5\right)dx=\frac{\pi}2[/math]
[math]\cdots[/math]
[math]\int_0^{\infty} \mathrm{sinc}\left(\frac x1\right)\cdot \mathrm{sinc} \left(\frac x3\right)\cdot \mathrm{sinc}\left(\frac x5\right)\cdots \mathrm{sinc}\left(\frac x{13}\right)dx=\frac{\pi}2[/math]
[math]\int_0^{\infty} \mathrm{sinc}\left(\frac x1\right)\cdot \mathrm{sinc} \left(\frac x3\right)\cdots \mathrm{sinc} \left(\frac x{15}\right)dx =\frac{467807924713440738696537864469}{
935615849440640907310521750000}\pi[/math]
>>7729535
>>7729546
I surrender, third try and can't get this math shit to work.
And I've been here for years.
>>7729549
Make more breaks into it. i.e. instead of
\frac{x}{y}
write
\frac {x} {y}
Then he doesn't break it himself
>>7729567
ok thanks for the advice!
>>7729547
why are the first seven terms true and the eight not?
>>7729590
http://schmid-werren.ch/hanspeter/publications/2014elemath.pdf
If you can sum it up for me in 20 words ill give you a dollar
>>7729616
thanks for the paper! let's try it:
the terms are nothing more than convolutions with rectangles that have the property to "decrease" when [math]\text{sinc}{\frac{x}{1}} + \text{sinc}{\frac{x}{3}} + ...[/math] exceeds 1
something like that?
In a noncommutative field, a nonzero polynomial may have infinitely many roots
>>7729738
What are you, French? Fields are commutative in the civilized world.
>>7729746
Actually yes lol, I thought anglos were the ones who used "field" as possibly noncommutative but I guess not
>>7729491
I fed this into wolfram alpha once to check it, and it came back good. I did not perform a hand computation (a.k.a. an actual verification).
such polynomials are called /sparse/ polynomials IIRC.
I once asked /sci/ what text this comes from, but I don't recall getting an answer. I imagine that this kind of prose is the type of thing one might find in grad school texts, wondering if anyone can opine on that. Sort of taunting the reader: "You think you're so smart. You think you know things. Well now we're gonna REALLY learn you."
Furthermore, I've seen this cutie >>7729547 a few times, must be meme material by now (Brower something... Borstein?). One thing I've gotten fairly good at is bookkeeping aka formatting TeX, so I'm going to re-write this guy's >>7729546 thing since that ugly failcode is triggering my autism, knock on wood. Let's also re-index the integrand, to shorten the code and remove ambiguity about what "dots" mean...
[math] \displaystyle \int\limits_{0}^{\infty} \prod\limits_{n=0}^{7} sinc \bigg( \frac{x}{2n+1} \bigg) dx = \frac{467807924713440738696537864469}{935615849440640907310521750000} \pi [/math]
Other things I've learned while writing /tex on sci, anon:
-The other poster is 100% correct that a best practice is to add spaces - especially between "math" tags.
-it's also a good idea to enclose both limits (on an integral, on a sum, or like my product) in their own curly braces - this unambiguously associates the two "textual" arguments with a particular form.
-I always like to declare "displaystyle" reflexively. This simply makes everything look better, sharper and more legible.
I see lots of these suggestions not being followed in your raw code (having lectured you, my version had better post correctly in this post)!
Edit: made and deleted one fail-post. I suspect the long strings of numerals now. This may be another fail-post... (spaced the final "pi" out just once more before this attempt)
Just to prove my point that I actually know what I'm talking about (we're in general principles territory now), The failcode from >>7729947 is being repeated verbatim in this post, with the single change that the long numeral strings are shortened (with the result of a false equation):
[math] \displaystyle \int\limits_{0}^{\infty} \prod\limits_{n=0}^{7} sinc \bigg( \frac{x}{2n+1} \bigg) dx = \frac{4}{9} \pi [/math]
>>7729960
And finally, the original long fraction, with its numeral strings will be repeated by itself, here. A failure to format in this case would pretty well diagnose anon's original problem, and show that 4chan's tex implementation just doesn't like very long numbers. A success in formatting would be... interesting.
[math] \displaystyle \frac{467807924713440738696537864469}{935615849440640907310521750000} [/math]
>>7729491
But are there an infinite number of examples of this?
>>7729968
funny. the TeX works in the preview function.
let's try an ugly hack:
[math] \displaystyle \frac{\text{467807924713440738696537864469}}{\text{935615849440640907310521750000}} [/math]
>>7729975
I don't know. Perhaps a trivial infinite family, based on a single example, could be constructed (left as an exercise).
However, the language at a mathworld page suggests that the research on these critters is only in its most rudimentary stages, and less than a hundred years old. The thrust of the stub/article is "we've found a few over here, and a few over there", but there is no surface-level language (without real digging) to suggest that a comprehensive description, let alone any broad statements such as infinitude about these critters exists.
There is apparently a lower bound on the degree of such polynomials, which is directly relevant to the OP.
>>7729968
> [math] \displaystyle \frac{ 467807924713440738696537864469 } { 935615849440640907310521750000 } [/math]
>>7729590
>>7729616
>>7729649
Yes. I have not looked at the middle anon's pdf yet, but wiki provides a helpful "why" to the first anon's obvious (and good) question which amounts to what the latter poster is suggesting. See wiki for treatment.
https://en.wikipedia.org/wiki/Borwein_integral
To answer the question, the reason why the first seven iterations are true, while the eighth is not, is (apparently) because of a basic change in the sum of certain terms in the products, which can be expressed succinctly in terms of two "consecutive" finite series (where the pattern breaks), with an important number in between:
[math] \displaystyle \sum\limits_{n=1}^{6} \frac{1}{2n+1} < 1 < \sum\limits_{n=1}^{7} \frac{1}{2n+1} [/math]
After thinking about this for a bit, I was reminded of ratios of convergence, Taylor series, and basic calc series coursework. Admittedly these instead treat instead of infinite series (as opposed to the above finite series, let alone products), but the point is that there are familiar analogies by which to come to grips with this, albeit a a bit tortured:
"p series", for example (later revealed as special cases of the zeta function) converge when p > 1 (otherwise the "pattern" of convergence breaks down):
https://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29#p-series
and likewise for versions of geometric series
https://en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence
And any other friendly calc examples you can cook up where beyond some number, an exponent, term, parameter etc "breaks down" once you cross some elementary boundary: 0,1,2 etc.
>>7730016
I'm so pleased to see it that I ain't even mad (that you did in fact show me up using my own advice). Now we know what has to happen, and if this doesn't go then I'll be mad (at myself):
[math] \displaystyle \int\limits_{0}^{\infty} \prod\limits_{n=0}^{7} \mathrm{sinc} \bigg( \frac{x}{2n+1} \bigg) dx = \frac{ 467807924713440738696537864469 } { 935615849440640907310521750000 } \pi [/math]
>>7729989
Is the primegap thing real? It sounds like a load of horseshit to me.
>>7729947
google fu!
>>7730134
Yitang Zhang's paper has been published and refereed by top number theorists.
I believe the prime gap now is even much smaller than what's in that image (due to Zhang), maybe even around 700 but I'd have to look again
>>7730473
And so it's now 246:
http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes
