Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: $$F_1=2+1\to prime$$ $$F_2=2^2+1\to prime$$ $$F_3=2^{2^2}+1\to prime$$ $$F_4=2^{2^{2^{2}}}+1\to ?$$ And so on. Amazingly, it has been found that $F_1$ through $F_{15}$ to […]

Here : https://sites.google.com/site/largenumbers/home/3-2/andre_joyce Saibian presents the largest number coined by Andre Joyce, googolplum. It should lie at the $f_{\omega+2}$-level in the fast growing hierarchy, but where exactly ? In other words : Between which tight bounds does googolplum lie ? Is it really at level $f_{\omega+2}$ ? If yes, what is the smallest number $n$, […]

TREE(3) is the famously absurdly large number that is the length of a longest list of rooted, 3-colored trees whose $i$th element has at most $i$ vertices, and for which no tree’s vertices can be mapped to the vertices of a subsequent tree preserving color and inf relationships. Some lower bounds of TREE(3) have been […]

Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\cdot\ln(n))-1)+\frac{n(\ln(\ln(n))-2)}{\ln(n)}$$ we get that $p_{10^{100}}$ is somewhere between $2.346977\cdot 10^{102}$ and $2.35698\cdot 10^{102}$ and approximately $2.3471\cdot 10^{102}$ , so it has $103$ digits. How many digits can we determine of the googol-th prime with […]

Thanks and with respect to the users of this site, I’ve succeeded in creating an Encryption/Decryption procedure for the RSA algorithm. I also implemented a Miller-Rabin probabilistic primality test. Now my question is much simpler, but my head really cannot make up an answer. Suppose I need a key of length 1024, and I’ve generated […]

Suppose, we have two numbers $a\uparrow \uparrow b$ and $c\uparrow \uparrow d$. To avoid trivial cases, suppose $a,b,c,d>2$ and $(a,b)\ne (c,d)$. Is there a quartupel $(a,b,c,d)$ with $a\uparrow\uparrow b=c\uparrow \uparrow d$ under the given conditions ? In general, $b<d$ implies $a\uparrow \uparrow b<c\uparrow\uparrow d$, but if $a$ is large enough, $a\uparrow\uparrow b$ will exceed $c\uparrow […]

Could it be at all possible to calculate, say, $2^{250000}$, which would obviously have to be written in standard notation? It seems impossible without running a program on a supercomputer to work it out.

Context: If you guys didn’t know, I’m running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to join). If you are entering, do note that I am accepting entries for quite a while (ignore all […]

Probably a proof (if any exist) that calls upon Knuth’s up-arrow notation or Busy Beaver.

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number of solutions? Are there ones with so large number of solutions that we cannot write any explicit upper bound for this […]

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