100 trillion digits of π were calculated in 2022 --- cloud.google.com/blog/products/compute/… ; $\sqrt 2$ is known to 10 trillion decimal places --- mathscareers.org.uk/how-well-do-you-know-the-square-root-of-2

10,000,000,001,000 decimal digits are known. numberworld.org/y-cruncher

Now your question made me wonder which kind of clients might be attracted to buy which kind of services if they are shown a large sequence of digits of $\sqrt{2}$.

Let us also agree that a billion is $10^{12}$, a trillion is $10^{18}$, a quadrillion is $10^{24}$, a quintillion is $10^{30}$, and so on.

@GHf, why would we agree to that, when everybody (where I come from) knows that a billion is $10^9$, a trillion is $10^{12}$, and so on?

@Jochen, I am targeting companies in the security and synthetic data fields. To get very large numbers of digits obtained very fast, and more random (thus more secure) than anything on the market. It's not just $\sqrt{2}$, but any quadratic irrational.

I’m not sure that counts as “random”…

@Zhen, if you don't know which quadratic irrational, and at what location you start in the digit sequence, for all purposes it is random. No one would be able to correctly predict the next 20,000 digits, given the previous 50,000 ones. Ready to bet $100k on that. Assume I start at an arbitrary location beyond the first trillion digits.

That can be made into a precise claim which the number theorists probably have an answer for. You might want to see what they say first.

@Zhen, the chance of winning my bet is lower than $2^{-20000}$. Assuming these numbers are all normal, a conjecture widely believed to be true (the numbers in questions passed all the statistical tests of randomness). Especially if you skip the first hundred thousand digits where randomness may not be as pure.

@GerryMyerson Just think about what "bi", "tri", quadr", and "quint" stand for in named numbers. And you will know the answer.

@VincentGranville Questions like this consume much more than single decades of single human lives, Vincent; I mean no personal offense to you in closing the question or my comment, only to offer due respect to the mathematics in play.

@GHf, given a language in which "cleave" can mean both "separate" and "adhere", "sanction" both "permit" and "penalize", "bolt" both "leave quickly" and "immobilize", "fast" both "moving rapidly" and "unmoving", the attempt to derive meaning through logic is unconvincing.

I disagree with the close votes, in particular with the reasoning that this were not about reseach level mathematics (and I thus voted to reopen). OP is working on (or plans to work on) an algorithmic problem and wants to apply this to create a pseudo random number generator based on analytic and statistic properties of certain real numbers. Implementing this most likely requires non-trivial mathematical methods (probably related to fast fourier transform algorithms and the like) and OP's goal is to break the currently known benchmark. [...]

[...] No offence intended towards anyone, but I think this certainly qualifies as mathematical research, unless one applies a rather narrow interpretation of the notion "mathematical research". Regarding the specific question asked by the OP, it seems to be a reference request (about state of the art regarding a very specific question). This also seems to be on-topic, since we have the "reference request" tag (which I added to the question) for such kind of questions.

@Vincent: That said, I believe some of the information that you provided in the comments should probably be edited into the question. Maybe this would also help the question to be received more positively?

@JochenGlueck If the question asked for algorithms suited for this problem, then I would be inclined to agree. However as written the question doesn't do that, it merely just asks for the highest known number. The original post also contains no mention of trying to produce pseudorandom sequences, only trying to "attract clients to buy services that I offer". The question can be edited to be suitable here but for the question as written the close votes were well justified.

To impress friends just say that the 12345678910111213141516171819-th digit of π is 4 and dare anyone to prove you wrong! Better yet, take 9 of your buddies and each claim a digit, whoever gets the client spreads the wealth...

@VincentGranville "If there is one person who made the biggest advances towards proving anything substantial regarding the normalcy of $\sqrt{2}$, that would be me." A strong claim indeed (especially when followed by accusing others of arrogance). What have you proved regarding the normalcy of $\sqrt{2}$?

@Carl, I haven't proved the normalcy in question. Few if any ever mentioned that the limiting proportion of 0/1 may not even exist. That's the case for a very large dense set of numbers (of zero Lebesgue measure). My goal is to prove something like this: in the following list of quadratic irrationals, there is at least one where $\lim \sup$ (for the proportion of 1) is strictly less than 100%. Such a basic result has never been proved. To put it is this way $\sqrt{2}$ is surrounded by plenty of very, very close neighbors (at distance 0) with no limit of uneven 0/1 distribution.

@Jochen: here is one of my applied papers on the topic (there is something wrong it it that was updated in the new version not yet published): github.com/VincentGranville/Main/blob/main/…

The number may well be normal, but that does not make it a cryptographically secure pseudorandom number generator. You know, the Champernowne constant is normal, but I hope everybody will agree that it would make a lousy stream cipher.

@VincentGranville Maybe my question was unclear; I am not asking what your goal is to prove, but rather what you have proved (if you have indeed done more than anyone else on the subject, then this should surely be easy to produce an example of?). I fail to see what mathematics is new in the paper you linked.

@Carl: "What have you proved regarding the normalcy of $\sqrt{2}$ ?" Digits of $\sqrt{2}$ and $\sqrt{3}$ are uncorrelated, using the empirical correlation metric. Or autocorrelations of any lag are zero for "good seeds" in the chaotic version of the logistic map (which is homomorphic to the dyadic map, thus the connection to binary digits). To summarize, I have essentially reduced the complexity of proving the normalcy to proving Collatz conjecture. If that one is now proved, would love to see the proof because it might be adapted to prove something about normalcy of $\sqrt{2}$.

@VincentGranville What is the "empirical correlation metric"? Do you have a link to these results?

@Wojowu: here's a link: github.com/VincentGranville/Main/blob/main/…. I don't post this stuff anymore because then I am accused of spamming. But when I don't post link to my own research I am accused of having not done any research.

@Carl: "What is the "empirical correlation metric?" - just the observed correlation between two finite sequences of $n$ numbers (the first $n$ digits) when you let $n\rightarrow\infty$. Proof is unpublished, but you can see a different version (not from me) at stats.stackexchange.com/questions/422354/…

@VincentGranville Alright, if one of your prime examples of your contributions to proving the normalcy of $\sqrt{2}$ is a StackExchange answer by another user, and your own work is all unpublished (except some extracts of basic Python code), then I think claiming that you have contributed more than anyone else to the subject is not only wrong, but also quite insulting.

@Emil, I suppose you are smart enough to know that comparing Champernowne vs. $\sqrt{2}$ is like comparing apples and oranges.

BTW, is the decimal expansion of $\sqrt{2}$ equal to the Champernowne sequence? (Apart from a few initial digits ofc.)

Carl definitely writes about BS sometimes, see e.g. page 3 on arxiv.org/pdf/2208.07145.pdf . (Sorry I felt the mood needs to be lightened.)

@Vincent maybe time to take a break from the conversation. Insinuating another users research is "bs", and comparing Scholar profiles to win a point is not going to help anyone. This question has an accepted answer, might it be worth moving on with your research, rather than beating a dead horse in the comments? :-)

@Carl in light of the general temperature in the comments, not the best choice of words. No one needs to 'win' the debate here, so I suggest dropping it :-)

I or another mod will clean up the comments here after a bit. MO has a reputation for unkindness, it behoves us all as regular users to try to turn that around, even if an epsilon.

Maybe for this and similar questions it would be more appropriate to ask how many digits have been printed on paper. (But I'm not sure we would want to the outside world to know that.)

@Dave: the rest of us? Nope you are talking about you and a clique of other folks, many with limited math experience and a narrow view of what research is. The two worst offenders in this thread appear to be the two with the least experience including none on this particular topic. If you don't like my content I assume you can turn me off. You would do me a favor.

if you believe those numbers are normal, then every string of 70 000 digits appears. i hope this bet gets offered

@Vincent if by 'Dave' you are meaning me, the system doesn't recognise that as my username, and I prefer people don't use that familiar appellation for me, especially strangers: my name is David, not Dave. I didn't say "the rest of us", I said "all of us", namely all regular MO users whose interactions here set the tone of discussion as an example to new users or drive-by readers. I'm going to shuffle this discussion into chat, because it's adding little to addressing the question, by this point, and is distracting from the mathematics. It would be a poor example at a public talk to bicker so.

@Zack, not only that, but they appear infinitely many times. I also defined the concept of strong normality, but the purpose of my question was not to delve into all the details.