cryptography – How are extremely large integers stored and implemented in programming languages?

MPI stands for Multiple Precision Integer. Multiple precision arithmetic is what you need when you work with integer types that go beyond the machine width $w$.

The basic idea is simple, you represent a large integer with multiple fixed-width words where the i-th word is the i-th “digit” in base B where $B = 2^w$.
For example, most current machines are 64-bit so the width $w$ is 64, so with a single word you can represent unsigned integers up $2^{64}-1$. To represent integers larger than $2^{64}-1$, say a 1024-bit integer as in your RSA example, you use $lceil{1024 / 64}rceil = 16$ words $a_0, a_1, a_2, ldots, a_{15}$. Then your integer $x$ of choice is encoded as
$$
x = a_0 + 2^{64} a_1 + 2^{2*64} a_2 + ldots + 2^{15*64} a_{15}.
$$

Note that this is essentially a $1024$-bit representation, the only difference is that the bits are grouped into blocks of size 64.

Operations like additions, multiplication etcetera are implemented by building on machine addition, multiplication and so on, but of course additional work is needed to take care of carries and the like. I am not sure what the Linux kernel is using, but in the GNU/Linux world a widely used multiple precision arithmetic library is the GMP.

google chrome – Write a C program to check whether two or more non-negative given integers have the same rightmost digit

My code pl. check and let me know correction.

#include<stdio.h>
int main(){
int x,y,z;
printf(“Enter the numbersn”);
scanf(“%d%d%d,&x,&y,&z”);
if (x % 10 == y % 10 || x % 10 == z % 10 || y % 10 == z % 10)
{
printf(“t”);
}
else {
printf(“False”);
}
return 0;
}

process scheduling – Reason that integers are used for priorities instead of float

Why are priorities always from a fixed set of integers? In operating systems priorities are integers typically between 1..MAX. But what is the reason that this is the case, when it would obviously be easier to place a new task between priority 2 and 3 with priority 2.5 instead? The way it is now, my scheduling can run out of priorities if my operating system has priorities between 1 and 10 and I need to schedule more than 10 different priorities or must re-arrange for a new task that should be between two subsequently scheduled tasks.

nt.number theory – If $a$, $b$, $c$ are three consecutive positive integers then show that $ab + c = s^n$ is not possible?

nt.number theory – If $a$, $b$, $c$ are three consecutive positive integers then show that $ab + c = s^n$ is not possible? – MathOverflow

nt.number theory – how many 0-subset-sums can a set of integers admit, none of whose 0-subset sums feature adjacent elements?

Consider an array of (possibly non-distinct) integers $a_0, ldots , a_{n – 1}$, where $n$ is even, and an additional integer $a$. Suppose that the $a_i$‘s $a$-subset sums lack adjacent elements, in the sense that for every subset ${i_0, ldots , i_{k – 1}} subset {0, ldots , n -1 }$ such that $sum_{j = 0}^{k – 1} a_{i_j} = a$, it also holds that ${2 j, 2 j + 1 } not subset {i_0, ldots , i_{k – 1}}$ for each $j in {0, ldots , frac{n}{2} – 1}$.

How many subsets ${i_0, ldots , i_{k – 1}} subset {0, ldots , n – 1}$ can there be such that $sum_{j = 0}^{k – 1} a_{i_j} = a$? Though the constraint on adjacent elements gives a theoretical maximum of $3^{frac{n}{2}}$, extensive empirical evidence shows that the actual optimum is $2^{frac{n}{2}}$. This latter maximum can clearly be attained, for example if $(a_0, a_1, a_2, a_3 ldots , a_{n – 2}, a_{n – 1}) := (1, 0, 1, 0, ldots , 1, 0)$ and $a := 0$. Thus the conjecture is that this is the best you can do. I think this should still work if the $a_i$ come from say $mathbb{Q}$, or any finite field of characteristic $p > n$.

This conjecture also be phrased in terms of boolean and linear algebra. Define the boolean function $f : {0, 1}^n rightarrow {0, 1}$, $f : (x_0, ldots , x_{n – 1}) mapsto (overline{x_0} vee overline{x_1}) wedge cdots wedge (overline{x_{n – 2}} vee overline{x_{n – 1}})$ (this is called the “Achilles heel” function in boolean logic). Then the conjecture is that even though $left| f^{-1}(1) right| = 3^{frac{n}{2}}$, any $mathbb{Z}$-submodule $A subset mathbb{Z}^n$ such that $A cap {0, 1}^n subset f^{-1}(1)$ actually satisfies $left| A cap {0, 1}^n right| leq 2^{frac{n}{2}}$, which is well under the a priori maximum.

linear algebra – Prove that $ frac{x}{y} + frac{y}{z} + frac{z}{x}=1 $ has no solutions in positive integers $ x,y,z $

Prove that $ frac{x}{y} + frac{y}{z} + frac{z}{x}=1 $ has no solutions in positive integers $ x,y,z $

I got this problem on brainly.in. They have no solutions about this.

I tried to solve it, here my attempt.

$$
begin{split}
frac{x}{y} + frac{y}{z} + frac{z}{x} &= 1 quad text{multiply by } xyz \
x^2z + y^2x + z^2y -xyz &= 0\
x left( zx+y^2 – xy right) + z^2 y &= 0
end{split}
$$

From here i need to prove $ xz + y^2 -xz geq 0 $ but i am failed to do so.

How can i prove it further?

products – Exact angle between two vectors with integers coordinates

I looking for two vectors in 2D, $u = (x, y)$, $v=(z,t)$ with integer coordinates ($x, y, z, t$ are integers) such that the angle between $u$ and $v$ is exact.

I use the formula
$$text{angle between } u, v = frac{u cdot v}{lvert u rvert cdot lvert v rvert},$$
where $cdot$ denotes scalar product ($u cdot v = x cdot z + y cdot t$) and $lvert cdot rvert$ is the modulus)

I think it is impossible to find an exact angle if coordinates are integers. But I cannot prove or disprove.

Find minimal spanning tree of graph with edge values from 1 to 5 integers

How can I find the minimal spanning tree of graph with edge values from 1 to 5 integers (no need to be unique) most effectively? I know I can use Kruskal algorithm, but how can I modify the algorithm to find it faster when I know there are edges with values only 1, 2, 3, 4 or 5? I cant figure it out how it could be faster when I know this limitation on edges.

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