It's ironic to see that 11MB file being called "gigantic" in this day and age; many web pages are now unfortunately much larger, and the page of the article itself transfers at least 1MB of data.

A thorough joke.

Google told me that, surprisingly, it's not even known that they are infinite!

I found this nice page on the Wagstaff Mersenne Conjecture:

https://primes.utm.edu/mersenne/heuristic.html

We eventually realized that no one really cared about the numbers we felt confident we could find--now Mersenne primes, though, people really care about those, but they're extraordinarily difficult to find. Always a cool day to see a new one.

PS: I remember scheduling Prime95 to run on lab computers at work (a nuclear engineering shop) after-hours, c. 1998.

https://www.mersenne.org/various/math.php#lucas-lehmer

https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality...

It makes me think that there should be a list of viable uses of blockchain, and another list of interesting problems that could be moulded into PoWs, and they should be paired up into blockchains.

Specifically,

a) Spend some coins to buy some other cryptocurrency

b) use all their secret primes creating a bunch of new blocks on a branch where they haven't spent their money. Since this is the longest branch now, all the other clients accept they haven't really spent it.

c) buy the other cryptocurrency again on this new branch.

Update: found some clues here, https://www.reddit.com/r/primecoin/comments/1i71ma/strange_b...

The idea is to combine the ideas behind Namecoin with Primecoin.

Also your concerns are unfounded: See Namecoin and Primecoin. I'm thinking these two ideas could be combined.

If it was real and explainable by science, we won't call them supernatural

whoa

PS: Atheist ;)

And: "Likely"? Objection, your honor. Assumes facts not in evidence.

https://en.m.wikipedia.org/wiki/Law_of_the_instrument

For example, if 2 were not prime, it would be impossible to represent the number 256 (or the number 123456) as a product of primes.

  >>> sympy.factorint(6)
  {2: 1, 3: 1}
  >>> sympy.factorint(1)
  {}

The fundamental theorem doesn't have to state that the prime factorization is nonempty.

It's true that some mathematicians have defined 1 as a prime number and there's nothing logically inconsistent about doing so, but it makes most theorems and formulas in number theory more complex and so this definition has fallen out of favor.

Edit: I think the Wikipedia article on the empty product gives some quite nice examples of the benefits of a closely related concept. https://en.wikipedia.org/wiki/Empty_product

Arithmetician (Ph.D. in number theory) here. This answer is completely correct.

As another reason why the factorization of 1 should be the empty set: suppose you have two positive integers m and n. Write S(m) and S(n) for their sets of prime divisors, counted with multiplicity.

Then S(mn) is the union of S(m) and S(n). We need S(1) to be the empty set to make this rule consistent.

Analogous to how log(1) is equal to 0.

The analogy is a good one, as zero is the additive identity (https://en.wikipedia.org/wiki/Additive_identity) of the integers, and one is the multiplicative one (https://en.m.wikipedia.org/wiki/1#Mathematics)

I'm not saying it's incorrect, but you have to explain that edge case away. At the same time allowing 1 to be prime would remove that problem given that e.g. 6 = 2^1 * 3 ^1 and 6 = 1^81 * 2^1 * 3^1 are the same unique factorisation because, well, monoids.

Edit: `s/fields/monoids`

Now you are the one squirming.

Cleanly accounting for edge cases isn't squirming, it's logical thinking.

What the sibling response to my own is correct. It's just the empty set.

Heh.

https://news.ycombinator.com/newsguidelines.html

Heck, mathematicians in general thought number theory was “useless” up until only around 40 years ago. Now we know that our world could not have developed this way without the creation of modern cryptography.

I think number theory serves as a great example for why basic research is so important to humanity.