Matrix of Primes

Computer Science Level 4

$\large {\begin{bmatrix} 3 & 5 \\ 2 & 7 \end{bmatrix}}$

Above is a $2 \times 2$ matrix having elements from the set of first $4$ prime numbers i-e $S= \left\{ 2,3,5,7 \right\}$ such that they may or may not repeat.

For how many ordered tuples $T(w,x,y,z)$ , where $w,x,y,z$ belongs to the set $S$ , determinant of the matrix $M=\begin{bmatrix} w & y \\ x & z \end{bmatrix}$ is also prime ?

This problem is a part of set Prime Crimes via Computer Science

The answer is 35.

3 solutions

Rushikesh Jogdand
Jul 1, 2016

s=[2,3,5,7]
def dt(ls):
    w,x,y,z=ls[0],ls[1],ls[2],ls[3]
    return w*z-x*y
from gmpy import is_prime
co=0
for a in s:
    for b in s:
        for c in s:
            for d in s:
                if is_prime(dt([a,b,c,d])) and dt([a,b,c,d])>0:              #negatives are not counted
                    co+=1
print(co)

Instead of bring lazy and using gmpy , I'd have just listed primes up to $43$ as largest determinant is $45$ .

展豪張
Mar 23, 2016

No solution...just brute force it :)

What if you followed the solution. How would you do it? :P

Zeeshan Ali - 5 years, 2 months ago

What do you mean by followed the solution?

展豪張 - 5 years, 2 months ago

What if you wanted to follow an algorithm? How would you do it? :)

Zeeshan Ali - 5 years, 2 months ago

@Zeeshan Ali – I'm using python here (pseudo-code) (maybe a stupid approach):
Prime_list=[2,3,5,7,...,43] (max $=7^2-2^2=45$ , max prime under $45$ is $43$ )
Count=0
Run $w,x,y,z$ through $\{2,3,5,7\}$ (four-layer for-loop)
if determinant in Prime_list, Count+=1
print(Count)
=D

展豪張 - 5 years, 2 months ago

@展豪張 – That's it.. you are good :)

Zeeshan Ali - 5 years, 2 months ago

@Zeeshan Ali – Thank you. This is a good question. Actually the whole set is challenging!

展豪張 - 5 years, 2 months ago

Arkajyoti Banerjee
Mar 2, 2017

#include<iostream>
using namespace std;

int comp(int &n)                //function for checking whether number is prime or composite
{
    int s=0;
    for(int i=1;i<=n;i++)
        {
            if(n%i==0)
                    s++;        //calculates no. of divisors
        }

    if(s==2)                    //divisible by 1 and the number itself
            return 1;
    else
        if(s>2)                  //divisible by more numbers other than 1 and the number itself
                return 0;
        else                    //elimination of 1 as it is not a prime number
                return 0;
}
main()
{
    int a[4],i=0,j=0,k=0,m=0,t=0,d;
    a[0]=2;                         //array
    a[1]=3;                         //of
    a[2]=5;                         //the
    a[3]=7;                         //4 primes

    for(i=0;i<4;i++)
        {
            for(j=0;j<4;j++)
                {
                    for(k=0;k<4;k++)
                        {
                            for(m=0;m<4;m++)
                                {
                                    d=a[i]*a[m] - a[j]*a[k]; //determinant
                                    if(comp(a[i]) && comp(a[j]) && comp(a[k]) && comp(a[m]) && comp(d))
                                        {
                                            //cout<<a[i]<<" "<<a[j]<<" "<<a[k]<<" "<<a[m]<<endl; //optional statement to print the tuplet
                                            t++;
                                        }
                                }
                        }
                }
        }

    cout<<"\nTotal = "<<t;
}

Matrix of Primes

This problem is a part of set Prime Crimes via Computer Science

The answer is 35.

3 solutions

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