What are Prime Numbers?
A prime number does not have any other factor besides one and itself. These whole numbers are greater than one. A factor is a whole number that you can divide with other numbers evenly. The list of prime numbers includes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on. Starting from 1, there are only 25 prime numbers to 100. When a whole number has more factors than two, you can call them composite numbers. We will not consider one as a composite or prime number. In other words, you can only divide a prime number with one and itself without the remainder. For instance, you can only divide 17 by one and by 17.
Some Important Facts about Prime Numbers
- 2 is the only even prime number. You can divide all other even numbers with 2
- You can divide a number by 3 if the sum of a number is a multiple of 3
- There isn’t any prime number that is greater than 5 and does not have the last digit as 5 – You can divide any number with 5 that ends in 5
- You cannot consider zero and one as a prime number
- Every number is a composite or a prime number, excluding zero and one: It means that any number that is not a composite number is a prime number and vice versa
If you want to prove that a number is a prime number, you need to divide it by 2. So if the result is a whole number, it isn’t a prime number. Whereas if the number is not a whole number, then you can divide it by other prime numbers such as 3, 5, 7, 11, and so on.
Determining if the Number is a Prime Number
You can use a computer to find if a large number is a prime or not. Since there’s no limitation on how large a number can be, proving huge numbers as a prime is an arduous task. Even if you use a supercomputer, limitations are endless. For instance, the largest number that we know to be a prime number, so far, has 24,862,048 digits.
Experts are trying to formulate different algorithms to find a way and find even the greatest prime numbers. For instance, consider “n” as the whole number, but we do not know if it is a composite or a prime number. To find if it is a prime, we will take ½ as the power of ‘n,’ or take its square root. Now, you can round this number to the next biggest number and denote that with “m.” We can find these quotients:
qm = n / m
q(m-1) = n / (m-1)
q(m-2) = n / (m-2)
q(m-3) = n / (m-3). . .
q3 = n / 3
q2 = n / 2
This concludes that “n” is a prime number if q is the above derivation.
Mersenne and Fermat Primes
A Mersenne prime is a number you can reduce to 2 n – 1. In this form, the ‘n’ is a prime number. Here are some of the first known “n” values that can produce Mersenne primes:
n = 2, n = 3, n = 5, n = 7, n = 13, n = 17, n = 19, n = 31, n = 61, and n = 89
Whereas a Fermat prime is a prime number and a Fermat number. The form of the Fermat number Fn is 2m + 1. In this form, m is the power of 2. It means that m = 2n. Furthermore, the n in this form is the integer.
Prime Numbers and Cryptography
Encryption will always include the fundamental rule. It will include:
- The algorithm
- The actual procedure
Both these components do not have any secrets, but the key does. You can use Prime numbers to create various keys. For instance, the reason why public/private key encryption is essential is that you can easily calculate products by choosing two random prime numbers. Though, you will find it challenging and time-consuming to find the two different prime numbers and create a larger product. The reason it can be difficult is that you only know the product.
You can take a popular example of public-key cryptography in Rivest-Shamir-Adleman or RSA. This states that you will always find prime numbers as unique. Numerous applications use the prime numbers by the Digital Signature Standard (DSS) and the Diffie-Hellmen.
Is 258000 a Prime Number
No, 258000 is not a prime number, but it is a composite. You can write 258000 as the product of the prime factors. Here are the prime factors:
258000 = 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 43
If you convert this into exponential notation, you will write it as:
258000 = 24 × 3 × 53 × 43
Conclusion
There are numerous historical questions about prime numbers yet to solve. For instance, Goldbach’s conjecture signifies that you can express every even number greater than 2 as the sum of two primes. Furthermore, it says that you can make infinite pairs of prime, adding one even number in between. These types of questions encourage mathematicians to perform further advancement in the field of number theory. You can use primes for various information technology tasks.
