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...