Mathematicians Discover Prime Conspiracy (2016)
quantamagazine.org
quantamagazine.org
Ok, I verified on 10 000 tosses, it works, but I cannot see why. Can somebody explain?
Edit: nevermind, it can be found in the previous discussion.
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?"
Number Theory is basically the exploration of the deep consequences of a set of (ultimately arbitrary but apparently obvious) axioms. The exploration of those consequences arises from the fact that they are ultimately too plentiful, ramified, and incompletely provable (per Godel’s Incompleteness Theorems & Turing’s Halting Problem).
Isn't that true for all fields of pure math?
Number theory uses a much smaller set of axioms.
It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.
And yet the man who gave us physics ( newton ) also gave us calculus. Rather calculus was invented to help describe and model physics and the natural world.
Number theory is actually an example of pure math having real world impacts despite its nature. It’s very much about studying math for maths sake - though there are also people interested in it for their own applications.
They’ve gone the brute force route with number theory because proving things in it is hard - in the sense that a hard number theory problem takes centuries to solve otherwise.
I'm not sure how to feel about that reminder of age.
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 think that's conjectural, but prime constellations are also conjectured to be a negligible fraction of primes as a whole, much as primes are a negligible fraction of integers (asymptotically). I think twin primes are conjectured to be distributed as n / (log n) ^2, while primes are n / (log n).
Besides, even if (p, p + 6) is significantly more common than (p, p + 2), if the final digit of p is uniformly distributed among 1, 3, 7, and 9, I don't think the statistics as a whole are affected.
Where you describe how to get your number, in relation to how it can be factored using prime numbers.
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.
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.
And while we still use Huffman’s algorithm, I wonder if advanced aliens may have discovered this technique for data compression.
This is more of a sci-fi fantasy algorithm, than anything actually mathematically sound.
So if you are compressing exceptionally large numbers, say, a whole video file, which is likely to contain very long factors, you could shave log2(log(N)) bits for each factor of length N. Seems like the most impractical compression algorithm imaginable.
The problem is that there can't be a way to compress every number: there aren't enough N-1 bit numbers to compress N bit numbers into (since the former set is only half as big as the latter).
Funny you should mention scifi because there's a story that got me interested in this topic [1] where some folks encode a large, important message into a compact algebraic expression like this. Turns out they're pissed at the recipients and the decimal expansion would be enormous so it would take them a while to decode it.
1. Fredrick Pohl, "Starburst", pp 56, 58, 59.
explanation for non-experts: "The Last Digit of Prime Numbers - Numberphile" https://www.youtube.com/watch?v=YVvfY_lFUZ8
In this regard, I don't see 2 as a special prime. Saying "2 is the only even prime" is the same as saying "2 is the only prime divisible by two", which, by definition, can be said about any other prime.
Anyway, just trying to react in tone to calling any number an “alien”. I think of numbers as being outside the realm of words like alien - I don’t even know what that would mean for a number. Hence my comment is hyperbole hinting at my love of mathematics. To me, it really is a beautiful area of study. Wish I could do it more.
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?
In that base, it can by 0, 1, or 2 mod 3 (for the final digit). Primes clearly aren't 0 mod 3, so all (not 3) primes end in 1 or 2. They found that primes "liked" to alternate between them.