How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring someone. I should have no problems producing 1 trillion, but almost sure I can't produce 1 quadrillion in any reasonable amount of time.


The goal is for marketing purposes and not to impress the mathematical community, but instead to attract clients to buy services that I offer. I'd like to also know about the digits of $\pi$, since producing a longer list (albeit for $\sqrt{2}$ or the golden ratio) would have a stronger impact.


**Update on 5/4/2023**


I am aware of what is in Wikipedia on this subject, and about the integer square root and how to compute it efficiently (for instance using the gmpy2 library). I develop PRNGs based on digits of millions of quadratic irrationals using new fast formulas, starting at arbitrary large locations, see [here](https://mltechniques.com/2022/12/13/military-grade-fast-random-number-generator-based-on-quadratic-irrationals/). If I claim that I can do better than what is in Wikipedia because I don't know the most recent computations, I could be accused of false advertising when stating that I have the longest sequence.


I want to avoid this, thus my question. The target customers will understand the random character of these digits, especially if offering a competition featuring 50k previous digits of one of these numbers starting at some location, and offer a large award (that I know no one will win) for correctly predicting the next 20k digits (with participants not knowing which starting location and which quadratic irrational I use).


I've made quite a bit of research on this topic, for instance - among others - proving that the digits of $\sqrt{2}$ and $\sqrt{3}$ are uncorrelated. The definition of correlation is in the comments.



How many digits of $\sqrt{2}$ are known to date?
In one sense, *all* of the base-2 digits (or, I guess, "bits") of $\sqrt{2}$ are known because we have a closed-form formula, according to the [OEIS](https://oeis.org/A004539): $$\begin{align} a(n) &= \frac{1}{2} - \frac{2\arctan(\cot(2^{-\frac{3}{2}+n}\pi))}{\pi} + \frac{\arctan(\cot(2^{-\frac{1}{2}+n}\cdot\pi))}{\pi} \\[10pt] &= \left\lfloor 2^{-\frac{1}{2}+n}\right\rfloor -2\left\lfloor 2^{-\frac{3}{2}+n}\right\rfloor \end{align}$$


**ETA**:


According to [Wikipedia](https://en.wikipedia.org/wiki/Square_root_of_2#Records_in_computation), the record number of digits for $\sqrt{2}$ as of this writing is $10^{13}+1000$. Further, [this website](http://www.numberworld.org/y-cruncher/records.html) purports to contain information regarding world records for the computation of various constants. I am unsure of how credible that site is but it came up in the citations on the Wikipedia page.


