If there are programs whose halting status is undecidable then there are clearly undecidable mathematical propositions. After all asking if a program will halt is a math problem.
Ok, forget about the coin flip. Lets say you have a box that contains 100000 molecules of a gas bouncing randomly. At any point in time there is a 50% chance that any given molecule will be on the left side of the box rather than the right side.
Would you say that if you looked and found all...
No.
Choose a compression algorithm. Then you can spend your entire life flipping coins and never find a 100000 bit sequence that will compress appreciably. The number of compressible sequences is so small that you will never see them by accident. And every bit you add to the sequence you want...
Meh, first there is no reason a computer can't generate random sequences. There is no requirement that a computer be fully deterministic. If its operation contains a random component then it can extract that random component and produce a random sequence.
If we are restricted to deterministic...
In QKD it only distributes a key so that you can communicate by a classical channel.
As long as the classical channel is public I don't see how man in the middle systems can break it. But you do need some kind of authentication system for the classical channel. That should not be hard with a...
I ask a very similar question here:
https://www.physicsforums.com/showthread.php?t=632458
What was your motivation for excluding 2?
I would be interested in what language and algorithm you used. I'm useing purebasic. I generated permutations of the prime list and constructed a gap by fitting...
And for N=7 we get a gap of 2*13=26 as expected and with N=8 we get 2*17=34 as expected.
But for N=9 we expect a maximum gap of 2*19=38. A gap of that size does exist but we have a larger gap with of 40 with an irregular structure...
OOPS! Sorry, my bad. That should be:
N=3--- P,2,3,2,5,2,P=2*3
N=4--- P,2,3,2,5,2,7,2,3,2,P=2*5
N=5--- P,2,5,2,3,2,7,2,11,2,3,2,5,2,P=2*7
To construct this gap we place the two largest primes in the middle with a two between:
....17,2,19......
Now going to the right we number the...
Now we are getting somewhere.
It is easy to show that the max gap for the first N prime numbers is at least two times the N-1th prime. We can always construct a gap of exactly this size. We construct the gap by listing the smallest prime a position in the gap is divisible by like this...
Why does it matter that they are not digits? why can't you have:
......1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14.......
for example? If you really need a sequence of digits rather than numbers you can just take out the commas...
Yes I think thats it but it really obscures what is going on. Just do a sieve of Eratosthenes using only the first N primes and find the largest relative prime gap that it leaves. But this method of finding the gaps is very slow.
There is another method using permutations of the prime list...
Ok, let me try again.
What is the biggest gap between consecutive numbers not divisible by two? Obviously it is two. 3 and 5 for example are consecutive numbers not divisible by two. They are not consecutive integers but they are consecutive numbers not divisible by two. They are also...
Yes I know that there are arbitrarily large prime gaps. I'm asking something different.
Again, given the first N primes, say 2,3,5,7 and 11, what is the largest gap between consecutive numbers not divisible by any of these? the answer for this (where N=5) is 14. I want a fast way to calculate...
Given the first N prime numbers what is the largest gap between consecutive numbers that are relatively prime to all of them? Anyone know of a fast algorithm for calculating this?
Look at the diagonal number that Cantor's argument can produce from this list of numbers. Which random number generator generates it? It can't be the first generator because the first digit is wrong. It can't be the second generator because the second digit is wrong. It can't be the third...
Only because the major philosophical issues in these branches have been settled long ago. For example few people doubt the existence of atoms today. But Ernst Mack, who influenced Einsteins development of relativity, didn't believe in atoms. I think there was a school of doubters clear up to...
Yes but Einsteins objection to QM was essentially philosophical in nature. He could not accept the loss of determinism. You cannot do science without addressing complex epistemological and ontological questions. Look at string theory and the objections to it for example.
Separating science...
Since the study of nature was once called natural philosophy then scientists are philosophers. Most philosophers don't deal with scientific method much but then most biologists don't deal with general relativity much. It is understandable that scientists would want to deny their history as...
Scientific method is a product of philosophy. Empiricism is a philosophical stance. The fact that most philosophers wander off into an intellectual waste land does not change this.
Actually I think what is important is understanding the ontological and epistemological consequences of Bell. Having predictions is nice and experimental confirmation of predictions is what science is. But it is the consequences that leave you gobsmacked.
Re: Joy Christian, "Disproof of Bell's Theorem"
Does everyone agree that anyone who claims to have developed a local realistic model for QM should be able to meet Sascha's Quantum Crackpot Randi Challenge...
bcrowell,
So this is old news already discredited? Thats a shame. But why is the story resurfacing on physorg news now. They really need better filters.
But what about Fermilab? They play with a neutrino beam much stronger than what the sun produces. Shouldn't they be able to see an effect quickly and cheaply?
I think this could be a game changer on many different levels.
First there is no known mechanism to allow neutrinos to affect radioactive decay. If it is happening we are probably seeing new physics. Always a good thing. It could also be some other dark particle in which case we are again...
Obama has issued loan guarantees to build two new nuke plants. Anyone know what kind of plants they will build? Are we really going to finally restart the American nuclear industry?