If I remember correctly, it has to be something that the person was actually capable of, otherwise the person just dies.
The fact that they entirely lucky that the series of digits indeed satisfies the mathematical condition we want is incidental but I believe abides by Death Note rules.
It's been a while since I watched Death Note, but isn't this brought up as a reason why Kira can't do, "writes down L's real name, then dies of a heart attack."?
It would be theoretically possible for someone to be lucky enough to guess L's real name, but apparently the note has a bullcrap detector or something where it eventually just says, "no, they wouldn't really know to do that."
Let me put this another way. I think the following would be considered a valid Death Note command.
"Person flips a coin 100 times in a row, and gets heads every time".
Now how is that different from
"Person runs an RNG digit generator 100 times in a row, and gets 9s every time".
Then
"Person runs an RNG digit generator 100 times in a row, the 100 digit number is exactly divisible by 123456".
Then
"Person runs an RNG digit generator 100 times in a row, 2^x-1 of that number is prime".
And if this is all iffy, let me propose an alternative. Let's assume (I'm not 100% sure of this) that we can verify large prime within 23 days. Heck, build a data center to do this if we have to.
"Person runs RNG to generate x such that 2^x-1 is prime. Uses machine and verifies that it is indeed prime. Dies of joy."
You can similarly use this to crack hashes, etc.
> "Person flips a coin 100 times in a row, and gets heads every time".
I'm pretty sure this would be interpreted as "The person has to flip coins until they have a 100 streak of only heads". This will obviously take a really long time, and the person might be driven insane before that. Similar things go for your other examples.
In a comment further up you wrote:
> in particular we know that the Death Note is able to "manipulate" luck
I don't recall that specifically. Even then, I would guess that this has it's bounds. I also have the feeling (not backed up by lore, but I think it would fit in the world), that bigger manipulations would take more energy from the respective shinigami, so while a shinigami might be able to manipulate luck enough to make someone win the lottery, finding prime numbers would be too draining.
I disagree. For example, Light has people die of getting run over a by truck. But the Death note does not interpret this is as "person repeatedly crosses the street until they get run over by a truck". Instead, events are (reasonably) manipulated such that a truck runs them over. Similarly, the coins would be manipulated to have 100 heads in a row.
>I also have the feeling (not backed up by lore, but I think it would fit in the world), that bigger manipulations would take more energy from the respective shinigami, so while a shinigami might be able to manipulate luck enough to make someone win the lottery, finding prime numbers would be too draining.
I wouldn't be surprised if this is true, but hey, what's a couple of Shinigami to save us some compute!
That being said... let's assume you're right. If the death note can manipulate odds to an arbitrary degree, you're thinking way too small right now. It would be much more efficient to write:
"X opens a laptop text editor, closes their eyes, and hits keys randomly. They open their eyes and discover they have accidentally written an algorithm for quickly factoring large numbers. They copy and paste this algorithm into an encrypted email to light.iskira@gmail.com, then die by jumping in front of a bus."
If X dies of a heart attack, you know your instructions didn't take and there's no algorithm for quickly factoring large numbers -- the problem was impossible. But then you can start modifying your query for, "an easy strategy for keeping cubits stable", or whatever you want to follow up with.
If I have a fast-track to get at almost any (possible) invention I can think of, I'm not going to waste my time factoring large numbers!
Right, what I feel might be the case is that the targeted output needs to at least be verifiable (hence see my second proposal). An argument could be made that unless the person has a way of verifying their action, there is no way to ensure that the desired outcome is achieved, so the instruction is invalid. (Separately, at least in your written example, it is possible for the person to be mistaken about the validity of their method!)
On the other hands, actions that are (machine) verifiable fall right within the scope of the Death Note (insofar as the verification instruments are also reliable, and verification happens within 23 days). I emphasize machine verifiable because the Death Note has various rules concerning other people, which I won't wade into. If "X starts his car and his car engine explodes" is valid, then so should "X keys a random string of inputs into a machine, machine returns True, then X dies."
So no we've limited the scope of DN-solvable problems to 23-day machine-verifiable problems. Now we just need to know what set of problems are 23-day machine-verifiable...
That's a good point.
> are 23-day machine-verifiable
Here's a followup question. Can a person be part of the machine?
For example, I might rephrase my earlier entry to be:
"X closes their eyes and types random keys, such that they form an algorithm for quickly factoring large primes. They email that algorithm to light.iskira@gmail.com and receive a confirmation that their algorithm is sound. At which point, they become so excited that they run into the street and are hit by a bus and killed."
My worry with that is that if there is some kind of force behind the death note, it might decide to just manipulate me instead of the person writing the algorithm. I don't want an evil genie scenario where I just accidentally write "yes, it works", instead of "no, it doesn't."
By extension, if we have a 23-day machine-verifiable problem, am I allowed to specify that the machine is not glitching or producing errors? In some cases, that might be more likely than finding the correct answer. Shinagami picking the most likely scenario that technically fits the criteria might be an effective way for them to reduce some abuse of the system.