## Prime Numbers Generating Formula using Matlab

A prime number is defined as any number larger than one that has no other divisor except for one and itself. An example is seven which can only be divided by one and itself. The numbers that do not meet these criteria are known as composite numbers. The divisors of a composite number could be other prime numbers. An example of a composite number is 35, which is divisible by 5 and 7. Both of them are prime numbers.

Identifying the prime numbers within a specified range of numbers might seem easy. In fact, they are taught at the primary level, and it’s practically easy for a child to understand the underlying concepts. But what has been intriguing most scientists are analyzing prime numbers on a large scale. That is up to millions.

## Is the number 1 a prime number?

For many years there was an argument as to whether one is a prime number or not. The confusion arises from the definition of a prime number, where it has been defined as any real number that can only be divided by one and itself. Consider the value one. It’s divisible by itself, and no other number can divide it. Evidently it’s not a composite number. Where can we classify it then?

The number one is considered as a non-prime number by all mathematicians. The number is a unique one as it cannot be divided by another number on the number line. Its multiplicative properties also make it unique. Remember a prime number involves a number that can be factored by another smaller number. For one, there is no other small number.

## What is the importance of prime numbers?

We have previously stated that finding the prime numbers in higher value numbers has been something that has interested many mathematicians over the years. But what is its importance?

Giant prime numbers have their use in data encryption. This is because there are few known data encryption algorithms that employ prime factorization.

They are cool. Perhaps you might not agree on this, but this is the general truth about these numbers. A proof to this is in the novel “contact”, where everything has been explained so rhetorically on why prime numbers are the building blocks of the number theory.

### History of prime numbers

The concept of prime number originated in the same place where mathematics is said to have originated. That is in the ancient Babylonian. The Babylonians were among the first to have become civilized. As a result, they had good knowledge of many mathematical concepts. Most of the mathematical concepts have originated from the Babylonians.

This civilized society was aware of the concept of prime numbers, but it’s the Greeks who were the first to study prime numbers systematically. Some of the notable contributions by the Greeks was through the famous mathematical genius, Euclid. He proposed that there is an infinite number of prime numbers which can be hard to find. That was over 2000 years ago. Serious attention to prime numbers was given in the 17th century by the Greeks, which lead to others supporting this research. As a result of the received attention, several formulas were proposed to find any prime numbers, but each had its differences. We shall look at them in this article.

### Formulas used to calculate the prime numbers

Finding the prime numbers for large numbers in the number line started a long time ago. Here are some of the formulas that you can apply to find prime numbers for the higher range of values, say you are told to find the prime numbers in the range of 0 to10000.

### Wilson’s theorem

This is a formula that is named after John Wilson, the man who proposed it. It was initially referred to as the Leibniz theorem. The formula is sufficient for finding prime numbers. The theorem states that any natural number that is greater than one, it’s referred to as a prime if and only if the product of all multiple values less than it is it’s multiple. Let’s an arbitrary value n, and we want to test for its primality, then by the Wilsons' theorem, the number is prime if (n-1)! +1 is divisible by n.

### Mills’s formula

This primality formula was developed by W.H. Mills in 1947. He wrote a paper that proposed that there exists a number such that dn = a3n. In this formula, n can take any number beginning from zero, and a is set of numbers which can be rational or irrational. Little is known about the constant n. By the Reiman hypothesis, then the smallest value of a can be said to be 1.3063778838630806904686144926… Despite being a good formula for finding prime numbers, it’s of no use in calculating the prime numbers if we do not know the prime numbers first.

### Wright’s formula

This formula bears a lot of similarity with the above formula, it can only be used to calculate prime numbers of known values. It of no use for giant prime numbers. It gives a constant α that has to adhere to some restrictions to find the prime numbers. However, the formula has its defects and this makes it unsuitable for finding prime numbers.

### Plouffe’s formula

This formula was introduced by Simon Plouffe. His formula is much the same as the Mills’ formula for primality it’s expressed as

{(aor)n}

Where the parenthesis {} implies that the number is to be rounded off to the nearest whole number.

## Formulas of prime numbers using Matlab

Matlab is a rich and resourceful mathematical computing software. In Matlab, there are a lot of readymade functions that one can use for any mathematical computation. However, there are a few instances where you have to create your own functions manually. But above all, it’s the software that can make your coding easier. The debugging process is easier. It enables you to correct the errors in the code by highlighting the errors. The advantages of Matlab are numerous.

Matlab comes with a function known as primes. As the name suggests, this formula will help you in deriving a list of prime numbers. Its general syntax is primes (n). So if you key in primes (100) on the Matlab console, it will return a vector list containing prime numbers less than n. Alternatively, you could develop your own custom made functions to find a prime number.

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This is a sample on Prime Numbers Generating Formula in Matlab. Here, the tutor is demonstrating the process of generating prime numbers. The generation of prime numbers is correlated to a formula that accepts integral values. Prime numbers are generated using the formula meeting certain constraints. The whole process has been automated using Matlab.

**SOLUTION: –**

clc

clear all

close all

i=1;

while 1 % This loop is for iterating over integers

x = i*i + i + 41;

flag=0;

% For checking whether x is prime or not

for j=2:floor(sqrt(x)) % Because if x is not prime, then it will have divisor between 2 and [sqrt(x)]

if(mod(x,j)==0)

flag=1;

break;

end

end

% If it is prime then stop the loop

if(flag == 1)

disp([‘Smallest value of n for which equation fails to produce prime number is ‘,num2str(i),’. For this equation generates ‘,num2str(x),’, which is not prime.’]);

break;

end

i=i+1; % Checking for next integer

end

% Checking whether next 100 numbers are prime or not

for k=i+1:i+100

x = k*k + k + 41;

flag=0;

% For checking whether x is prime or not

for j=2:floor(sqrt(x)) % Because if x is not prime, then it will have divisor between 2 and [sqrt(x)]

if(mod(x,j)==0)

flag=1;

break;

end

end

% If it is not prime then stop the loop

if(flag==1)

disp([‘No it does not generate prime for n=’,num2str(k),’. For this equation generates ‘,num2str(x),’, which is not prime.’]);

break;

end

end

% If all of next 100 numbers are prime

if(flag==0)

disp([‘It generates prime numbers for next 100 values of n.’]);

end