3
votes
0answers
65 views

Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
0
votes
1answer
29 views

Calculation of $f_{\omega^3}(2)$ in the fast growing hierarchy

How is the number $$\large f_{\omega^3}(2)$$ in the fast growing hierarchy calculated ? My only idea is to convert to $$\large f_{\omega^2 2}(2)$$ but now I have no idea how to continue.
0
votes
0answers
37 views

Monotony of the function $a\uparrow ^b c$ in all the arguments

I want to prove the following monotony properties of hyper-operations : 1) $ a \uparrow^b c < (a+1) \uparrow^b c$ for all $a,b,c \ge 1$ 2) $ a \uparrow^b c < a \uparrow^b (c+1)$ for all $a\ge ...
3
votes
1answer
110 views

Proof of the inequality $2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$

I tried to prove the inequality $$2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$$ for all natural numbers $n\ge 1$ For n = 1 , the claim is true because of 16 < 27 < 32. The left ...
2
votes
2answers
87 views

For which n does the inequality $2 \uparrow^{n+1}n > 3\uparrow^n 3 +2$ hold?

For which n does the following inequality hold ? $$2 \uparrow^{n+1}n > 3\uparrow^n 3 + 2$$ where $\uparrow$ stands for knuth's up-arrow notation. I need this inequality to prove that ...
2
votes
1answer
141 views

Definition of the function $f_{\epsilon_0}$ in the fast-growing hierachy

I found an article where the growth of $$f_{\epsilon_0}$$ in the fast-growing-hierachy is described, but the text is very long and difficult to understand. Is ...
1
vote
0answers
70 views

Tim Chow's proof that the moser-number is much smaller than grahams number

The link here shows a proof from Tim Chow that the moser-number is much smaller than grahams number. I do not understand the inequality 3^^...^^3 (3^^^^^3×2-1 arrows) << G 2 What does G 2 ...
0
votes
0answers
45 views

Bounds for $(10 \uparrow \uparrow 257) \uparrow \uparrow \uparrow (10 \uparrow \uparrow 257)$

A lower bound of 2[6] (Steinhaus-Moser-Notation) is $$ M:= (10 \uparrow \uparrow 257) \uparrow \uparrow \uparrow (10 \uparrow \uparrow 257)$$ I would like to bound M in the following way : $$10 ...
0
votes
0answers
24 views

Bounds for a sequence and the steinhaus-moser-numbers

Let $(a_k)$ be the following sequence $$a_1=n\ ,\ a_{n+1}=a_n^{a_n}\ for\ all\ n \ge 1$$ It is easy to show $a_k > n\uparrow \uparrow k$ for all $k > 2$. But is it true that $a_k < n ...
0
votes
0answers
35 views

monotony for conway chains

Let X and Y be chains, m and n be natural numbers with 0 < m < n. Is it always true that $$X \rightarrow m \rightarrow Y < X \rightarrow n \rightarrow Y$$ ? I need this to bound the ...
3
votes
0answers
251 views

Magnitude of $10 \uparrow \uparrow \uparrow 10$

Is there any way to understand the magnitude of $$10 \uparrow \uparrow \uparrow 10$$ ? I know that the number can be constructed as follows : $$M_1 := 10$$ $$M_2 := 10^{10^{10^{...10}}}$$ ...
1
vote
1answer
110 views

What are the last 20 digits of mega?

What are the last 20 digts of the number mega, which is "2 in a pentagon" in steinhaus-moser-notation ? In contrary to power towers or tetration, the ending digits are not stable. I found out that ...
2
votes
0answers
69 views

First digits of extremely large numbers (Generalization of “first digits of Graham's number”)

I found a question about the first digits of Graham's number and would like to generalize it : We want the first n digits of the number $a\uparrow^b c$. Which method is the most effective to do ...
1
vote
3answers
138 views

What is the tens digit of $3^{100}$?

Is there a general formula to calculate the n-th digit of any big number?
2
votes
0answers
60 views

Improving Montgomery product

I am reading the paper "A Cryptographic Library for the Motorola DSP56000" (http://link.springer.com/content/pdf/10.1007%2F3-540-46877-3_21.pdf) which describes a trick to speed-up calculation of the ...
1
vote
1answer
146 views

Find the largest divisor of an integer $b$.

I want to find out an efficient method to calculate the largest divisor of a very big integer $b$ which can be up to $\large 2^{1000}$. That is, I want to find out an integer $a < b$, such that ...
23
votes
5answers
460 views

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 ...
3
votes
3answers
223 views

Comparing $\large 3^{3^{3^3}}$, googol, googolplex

How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$). Thanks much in advance!!!
2
votes
1answer
509 views

How to calculate modulo of large integer (number having 25000 digits)

I'm looking for solution to a problem to calculate modulo of very large number that can contain 25000 digits or less (n) with 10 digit number (m). ( n % m ) ? Pointer to appropriate theory resource ...
4
votes
1answer
107 views

Are there any secure ciphers you can use without a computer?

I have some kids that like encryption schemes such as the Caesar cipher and the Vigenère cipher. I would like to teach them something that's not easily breakable by todays maths and computers, but I ...
4
votes
2answers
225 views

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. ...
52
votes
6answers
4k views

Is it possible to represent every huge number in abbreviated form?

Consider the following expression. 16313107343153908912074032799466965289077771751767944648966669091376847859711382649033004075188224 This is a 98 decimal digit number. This can be represented as ...