Articles of big numbers

How does one prove that $(2\uparrow\uparrow16)+1$ is composite?

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 […]

Where does googolplum lie in the fast growing hierarchy?

Here : 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 $n$, […]

explicit upper bound of TREE(3)

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 […]

How many digits of the googol-th prime can we calculate (or were calculated)?

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 […]

RSA: Creating a key of desired length

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 […]

Can different tetrations have the same value?

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 […]

Is there a way to calculate absurdly high powers?

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.

Could someone tell me how large this number is?

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 […]

What is the biggest number ever used in a mathematical proof?

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

Examples of Diophantine equations with a large finite number of solutions

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 […]