# Questions tagged [big-numbers]

For questions relating to the computation, estimation and properties of extremely large finite quantities that are not usually used in mainstream mathematics. This is not for questions that just have large numbers; the fact that a number is very large has to affect the question.

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### Largest known prime maximizing the relative digitsum?

A prime number with a given number of digits (except $1$) has the maximum possible digitsum (in base $10$) , if it has only digits nine except one digit which is $8$. A large (probable) prime of this ...
72 views

### Largest known prime with digit sum $4$?

According to OEIS , the number $$10^{509546}+3$$ is a (I guess probable) prime with digit sum $4$ in base $10$. Is it the largest known such prime ? What is the largest known such prime if it must ...
1 vote
148 views
+50

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### Calculate the n-th number of a power with huge exponent

There are several questions asked (e.g. 1, 2 or 3) on the last digit of numbers like $7^{355}$ or $237^{222222212202237}$. My question is, if there is any efficient method to calculate the n-th digit ...
1 vote
142 views

### Is the number of digits in Graham's number greater than the number of protons in universe (~10^122)?

I am wondering, is the number of digits in Graham's number greater than the number of protons in the known universe (~10^122)? Or is there some other 'big' lower bound to the number of digits in G64?
1 vote
87 views

### Probability of a big number that is composed of two special numbers to be multiple of 19

A big number, which is 117179 digits in length, was constructed by concatenation of only two special numbers, which are 40 and 8 (both 40 and 8 must be used). There are around 62870 of any of those ...
103 views

### How to convert an integer to a Knuth notation?

The Knuth notation enables to write very large number in a compact way. I however cannot find the way to do the reverse calculation.For instance, given the number $123456789$ , how can it be written ...
108 views

363 views

### Random search for very big Collatz conjecture counter-examples

I know that exhaustive search was done to test numbers up to 2^68. This seems like a big number but when looking at Collatz function as a Turing machine manipulating some input bit sequence, only ...
1 vote
48 views

### Birthday problem but with $2^{128}$ different days in the year [duplicate]

I am trying to calculate how many randomly generated ids I need to produce for there to be a 1% probability I get a duplicate id. There are $2^{128}$ possible ids. I understand this is just the ...
221 views

### Examples of rational numbers with large denominators appearing unexpectedly. [duplicate]

I am looking for examples of rational numbers with large denominators that pop up in questions in which there is a-priori no large numbers involved or no obvious reason why the denominator would be ...
216 views

### Most efficient way to square modulo a Mersenne prime

I realise this question is somewhere in between Math StackExchange and StackOverflow. So forgive me if this is too much of a practical question, it would probably too theoretical elsewhere. I am ...
1 vote
41 views

### What is the equivalent of breaking a n bits key with “winnig lottery every minute x times in a row”?

I was wondering how to give a hint to someone to imagine the size of a 256 bits key speaking about Bitcoin (even if private key in Bitcoin are 160 bits if I remember) and not with the number of atoms ...
377 views

### What is the largest finite number you can make using no more than 6 characters?

(I am new to the Math Stack Exchange community so tell me if this question is not allowed, however, I checked on Meta first.) (Also, I don't really know which tags I should assign this question, so ...
109 views

### A confusion about the TREE function.

I know, from Kruskal's tree theorem, that the sequence of trees mentioned in the TREE function cannot be infinite. However, why can't the sequence of trees get arbitarily large, so that there is no ...
1k views

### When does Busy Beaver surpass TREE(3)?

I saw a question which asked, "When does Busy Beaver surpass TREE(n)?" I am asking a somewhat different question, about a specific value of TREE. I know that TREE(3) is an unimaginably vast ...
1 vote
120 views

### First value of the second order busy beaver function?

There is a bit of talk about the busy beaver function. It was asked whether or not it is possible to make a function that grows faster than the normal busy beaver function. One way to do this is to ...
1 vote
926 views

### A confusion regarding Rayo's number and Busy Beaver function.

I am slightly confused about the definition of the Rayo function and Rayo's number, and how it relates to the Busy Beaver function. I know that ZFC can't pin down the precise value of even $BB(7918)$. ...
1 vote
346 views

### Linearizing the product of a binary and a continuous variable

I have an MIP optimization problem that has a constraint $p\geq xy$, where $x$ is a binary variable, $p$ and $y$ are non-negative continuous variables. I tried the Big-M method. However, the upper ...
322 views

### prime number (a form like Mersenne primes)

I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a ...
174 views

### Comparing power towers of $2$s and $3s$

Let $x=[x_1,x_2,...,x_n]$ be a finite list of positive real numbers, and define $\tau x$ as the power tower formed by these numbers. The function $\tau$ can be recursively defined by the following two ...
123 views

### What is the chance that a number $P$ is prime if it's not divisible by any number less than $x$?

I am trying to check if a very big number ($>10^{10,000,000}$) is possibly prime. I have written a computer program to check if the number has any smallish (less than like $600,000,000$) factors......
101 views

### calculating Modulus on Massive numbers [duplicate]

In the case I have two numbers large enough to justify using scientific notation twice $A \times 10^{B \times 10^C}$ or $Ae+Be+C$ How would I calculate Modulo without taking the numbers or any part of ...
### Given that $2017$ is prime, how do I prove this statement?
I'm asked to prove the following statement: Let $N=(1008!)^2+1$. Prove that $N$ is divisible by $2017$. (Hint: $2017$ is prime.) I don't know how to go about proving this statement, since there seems ...
### How to get the last $n$ digits of Ackermann function?
The Ackerman function is defined as follows: $$A(m,n)= \begin{cases} n+1,& m= 1\\ A(m-1,1), & m>0, n=0\\ A(m-1, A(m,n-1)), &m,n>0 \end{cases}$$ Is it possible to get the last ...