Being special is pretty, and cool because you have qualities that the majority do not have. This is why prime numbers are special and you would be knowing all about their uniqueness hereafter.
What is the explanation of prime numbers?
Prime numbers are natural numbers greater than \(1\) which can only be divided by itself and one which is all odd numbers except \(2\).
This means that a prime number has no factor other than it and \(1\).
Really cool right Because nothing gets to divide or share it into smaller whole numbers except \(1\) and itself.
Generally, integers greater than \(1\) are divided into prime numbers and composite numbers. Numbers greater than \(1\) which are not prime numbers are known as composite numbers.
Also, prime numbers are positive whole numbers (positive Integers), this means that negative numbers are not classified as prime numbers.
What are even prime numbers?
Even prime numbers are prime numbers which are even numbers. The only even prime number is \(2\) because \(2\) is a factor of all existing even numbers. \(2\) is also regarded as the smallest prime Number.
However, the largest current prime Number is \(2^{82589933} - 1\).
Meanwhile, there are other categories of prime numbers which would be discussed below.
Twin prime numbers
These are two prime numbers that have just a single composite number in between them. For example, \(3\) and \(5\) are twin prime numbers because they only have \(4\) between them whereas \(7\) and \(11\), are not twin prime numbers because between \(7\) and \(11\) you have three composite numbers such as \(8\), \(9\) and \(10\).
Repdigit prime numbers
These are prime numbers that are repdigit. A repdigit number is a number that is two or more digits and has the same digits repeated for example, \(11\), \(22\), \(33\), \(777\), \(5555555\), \(444\), etc. The only repdigit that is a prime number is \(11\).
Twist prime numbers
This is a pair of two digits prime numbers which are the same when reversed. For example, there are four twist prime numbers such as; \(13\) and \(31\), \(17\) and \(71\), \(37\) and \(73\), and \(79\) and \(97\).
What are the properties of prime numbers?
Prime numbers have the following properties:
a) They are all indivisible by any number except \(1\) and itself.
b) They are all greater than \(1\) with \(2\) as the smallest of them all.
c) They are coprimes of each other - this means that \(1\) is the only factor for any two prime numbers are factors of each other. For example, \(3\) is not a factor of \(5\).
Coprime numbers are two numbers that have no common factors except \(1\). For instance \(4\) and \(9\) are a coprime pair because \(1\) is the only common factor between both.
d) They are no repdigits amongst prime numbers except \(11\).
e) Any positive number equal to or greater than \(3\) is a sum of two similar or different prime numbers. For example, \(50\) is a sum of prime numbers \(37\) and \(13\); similarly, \(99\) is a sum of prime numbers \(97\) and \(2\).
f) All composite numbers can be expressed (factorized) as a product of prime numbers which are called prime factors. For example, \(26\) can be expressed as:
$$26=2\times 13$$
Likewise, \(54\) can be expressed as:
$$54=2\times 3\times 3\times 3$$
g) All prime numbers are odd numbers except \(2\) because \(2\) is a prime factor of all even numbers.
What is the formula for prime numbers?
There are two formulas commonly used in deriving prime numbers.
The first formula of prime numbers
The first formula used is:
$$\text{Prime number}=6n\pm 1$$
Where \(n\) is \(1, 2, 3, 4,...,n\). Meanwhile, this formula only works when the value is not a multiple of a prime number. If the value of \(n\) is a multiple of prime number, for example, when \(n\) is \(4\) we get,
$$(6\times 4)+1=25$$
\(25\) is not a prime number, as has a factor \(5\) other than \(1\) and \(25\).
Consequently, when applying this formula, do well to confirm if the value is indeed a multiple of other prime numbers.
Using the first formula, find prime numbers for \(n\) values of \(2\), \(3\), and \(9\) and confirm that they are prime numbers.
Solution:
Using the first formula where \(n\) is \(2\)
$$\text{Prime number}=6n\pm 1$$
$$n=2$$
$$(6\times 2)+1=13$$
$$(6\times 2)-1=11$$
When \(n\) is \(2\), we get \(11\) and \(13\), both are prime numbers.
Using the formula where \(n\) is \(3\)
$$\text{Prime number}=6n\pm 1$$
$$n=3$$
$$(6\times 3)+1=19$$
$$(6\times 3)-1=17$$
When \(n\) is \(3\), we get \(19\) and \(17\), both are prime numbers.
Using the formula where \(n\) is \(9\)
$$\text{Prime number}=6n\pm 1$$
$$n=9$$
$$(6\times 9)+1=55$$
$$(6\times 9)-1=53$$
When \(n\) is \(9\), we get \(55\) and \(53\). \(53\) is a prime number but \(55\) is not because \(11\) and \(5\) are prime factors of \(55\).
Second formula of prime numbers
The second formula used is applicable to only prime numbers greater than \(40\). The formula used is
$$\text{Prime number}=n^2+n+41$$
Where \(n\) is \(0, 1, 2, 3, 4,...,n\).
For \(n\) values of \(1\) and \(3\), find the prime numbers using the second formula of prime numbers.
Solution:
Using the formula when \(n\) is \(1\)
$$\text{Prime number}=n^2+n+41$$
$$n=1$$
$$1^2+1+41=43$$
Using the formula when \(n\) is \(3\)
$$\text{Prime number}=n^2+n+41$$
$$n=3$$
$$3^2+3+41=53$$
Examples of prime numbers
Between the numbers \(1\) to \(20\), what are the following present
a. Prime numbers.
b. A twist prime numbers.
c. Repdigit prime numbers.
d. Twin prime numbers.
Solution:
a) Using the formula
$$6n\pm 1$$
We would be able to find prime numbers such as \(5, 7, 11, 13, 17\), and \(19\). Meanwhile, the range is from \(1\) to \(20\). We know that \(2\) and \(3\) are prime numbers but the formula does not account for prime numbers less than \(5\). Thus the prime numbers from \(1\) to \(20\) are; \(2, 3, 5, 7, 11, 13, 17\), and \(19\). With this information, we can find the answer to the remaining questions.
b) Twixt prime numbers: The twixt prime numbers between \(1\) and \(20\) are \(13\) and \(17\) because when the digits of both are reversed you get prime numbers such as \(31\) and \(71\) respectively.
c) Repdigit prime numbers: The only repdigit prime number between \(1\) and \(20\) is \(11\).
d) Twin prime numbers: The twin prime numbers between \(1\) and \(20\) are; \(3\) and \(5\), \(5\) and \(7\), \(11\) and \(13\), and \(17\) and \(19\).
Prime Numbers - Key takeaways
- Prime numbers are natural numbers greater than \(1\) which can only be divided by itself and one which is all odd numbers except \(2\).
- Even prime numbers are prime numbers which are even numbers with \(2\) as the only even prime number.
- There are several properties of prime numbers.
- Two formulas are often used in calculating prime numbers such as:$$6n\pm 1$$where \(n\) is \(1, 2, 3, 4,...,n\)and$$n^2+n+41$$where \(n\) is \(0, 1, 2, 3,...,n\).
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