> If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.<p>Ok, I verified on 10 000 tosses, it works, but I cannot see why. Can somebody explain?<p>Edit: nevermind, it can be found in the previous discussion.
I find number theory to be closer to a natural, investigative science akin to Physics rather than pure math such as calculus.<p>IANAM, but I doubt there was ever a modern calculus theorem that was hinted by a brute force search.
One can argue that Calculus is "made up" and thus research in this field is finding deeper logical consequences for the initial axioms, while natural numbers just "have properties" and research is Number Theory is trying to uncover these properties by brute force and later trying to prove them, arguably quite unsuccessfully so far.
It's an interesting tidbit in the philosophical question of "Does math exist only in our head?"
Abstract:
While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.<p>- <a href="https://arxiv.org/abs/1603.03720" rel="nofollow">https://arxiv.org/abs/1603.03720</a>
Discussed at the time: <a href="https://news.ycombinator.com/item?id=11282480" rel="nofollow">https://news.ycombinator.com/item?id=11282480</a>
>This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. Most mathematicians would have assumed, Granville and Ono agreed, that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5).<p>This seems like an odd assumption to me. Surely sexy primes are more common than twin primes, so at least for primes that are near each other there should be a higher probability for certain sequences of final digits. This is obviously not proof in itself, but it would certainly make me hesitate to assume there is an equal probability in the general case.
I always wondered if prime numbers can be used as the basis of a compression algorithm.<p>Where you describe how to get your number, in relation to how it can be factored using prime numbers.<p>You cut up your data into chunks, and then find the prime factors of each chunk. Now you have a list of arithmetic descriptions to describe your data, and you can use Huffman to compress the descriptions.<p>Of course, this would be computationally intensive, for both the compressor, and the extractor, but it may save on the payload size, to allow for efficient data transfer.<p>And while we still use Huffman’s algorithm, I wonder if advanced aliens may have discovered this technique for data compression.<p>This is more of a sci-fi fantasy algorithm, than anything actually mathematically sound.
paper: "Unexpected biases in the distribution of consecutive primes" <a href="http://arxiv.org/abs/1603.03720" rel="nofollow">http://arxiv.org/abs/1603.03720</a><p>explanation for non-experts: "The Last Digit of Prime Numbers - Numberphile" <a href="https://www.youtube.com/watch?v=YVvfY_lFUZ8" rel="nofollow">https://www.youtube.com/watch?v=YVvfY_lFUZ8</a>
Classic example of following definitions or illustrious assumptions too closely... another one is an obvious one, and historically has been challenged a few times: the beginning of the prime sequence 'should be' 1, 3, 5.. very clean, very coherent, but it got stuck in definitions limbo and now we got no 1 and the alien 2, that has to be excepted in most algorithms..eh...
I wonder what the implications are for cryptography - given the heavy reliance on 'random' prime numbers<p>but one would have to delve into this - and from the top of my head there are no algorithms that make you pick 'two consecutive' primes - so the implications - if any - aren't obvious - anyone here has some more light or other angles to shed into this?
Not sure I understand the significance of a last digit of 9 for a prime implying the next prime is likelier to end with a 1 and not a 9. After all, base 10 is arbitrary. Can someone explain whether similar such "conspiracies" occur for all bases?
I wonder if it's a some extension or variation on Benford's Law [0].<p>[0] <a href="https://en.wikipedia.org/wiki/Benford%27s_law" rel="nofollow">https://en.wikipedia.org/wiki/Benford%27s_law</a>