A prime number is a natural number greater than 1 that can only be divided by itself and 1. Prime numbers are significant in many branches of mathematics, including as roots of polynomials, large odd-integer factorization, and cryptography. The study of prime numbers is part of number theory.
In this article, I am going to talk about prime numbers briefly and describe their significance in mathematics – both for students and professionals. I hope that after reading this article you will be able to confidently answer the question: What is a prime number?
Prime Numbers
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is a composite because it is the product of two numbers (2 × 3) that are both smaller than 6.
The first few prime numbers are 2, 3, 5, 7, and 11′
2 is the smallest prime number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Numbers can be tested for primality and also decomposed into their prime constituents.
A prime number is a natural number that has exactly two distinct natural numbers as its divisors: 1 and itself.
An example of a prime number is 13, which can be divided by one and itself only (and not 3, 5, or 7).
Primes are central theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. This theorem requires excluding 1 as a prime.
Another important consequence of this fact is that every integer has an infinite number of factors (itself, and all its positive divisors). For example, there are infinitely many factors for 6:
6 = 2 x 3 = 3 x 2 = 4 x 2 = 5 x 2 = 7 x 1
In contrast, non-prime numbers only have finitely many factors; thus any composite number has an infinite number of prime factorizations, but every single prime factorization will always be unique in some sense (see below). Conversely, while it may seem that there must be at least two distinct ways to write any given natural number as two or more prime numbers multiplied together (since you could write down lots more than just one way), this is not the case. A counter-example would look like this:
12 = 5^2 + 7^1
Why Are Prime Numbers Important?
Primes are also routinely used in cryptography. There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modeled.
A natural number greater than 1 that is not a prime number is called a composite number. The set of all prime numbers is denoted by N* or [P]. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17 but there are an infinite number of them after this point since they can be generated by adding 2 to any other number (the next one being 19).
Prime factors have been used in cryptography since ancient times because it is difficult to factorize large composite numbers into their constituent primes when computers were not available for this task (which has made factoring much more difficult).
Steps to Determine If a Number is Prime
The Sieve of Eratosthenes is a simple and efficient algorithm for finding prime numbers. It works by starting with the number 2, and marking it as composite (i.e., not prime). Then, starting at 3, it marks every multiple of 3 greater than 1 as composite (i.e., not prime), and every multiple of 2 less than or equal to its square root as prime. The algorithm terminates when no unmarked numbers remain.
The Sieve of Eratosthenes can be used to determine if a number is prime using the following steps:
- Start with 2 as an initial candidate;
- Test whether any of its multiples are divisible by 2;
- Test whether any of its multiples are divisible by 3;
- Repeat this process until all multiples have been tested and either found divisible or marked as composite;
- If there are no multiples left unmarked after step 4, then 2 itself must be prime since it has no known factors other than 1 and itself;
Some Important Facts About Prime Number
Prime numbers are whole numbers that can be divided by only one and itself. For example, 2, 3, 5, and 7 are prime numbers.
Here are some other facts about prime numbers:
- Prime numbers are called “prime” because they cannot be divided by any number other than 1 and itself.
- The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
- The largest known prime number is 2742582667366958632788… with more than 22 million digits!
- There are an infinite number of primes, but if you count them one at a time it would take billions of years for you to count past 10 million!
- All natural numbers greater than one can be written as a product of one or more primes.
- You can find many primes hidden inside Pascal’s triangle
List of Prime numbers up to 100
Here’s a list of all prime numbers from 1 through 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
There are 25 prime numbers below 100.
List of Prime Numbers between 1 and 200
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
There are 46 prime numbers that fall between 1 and 200.
List of Prime Numbers Between 1 and 1,000
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.
Mersenne and Fermat primes
Mersenne primes and Fermat primes are special types of prime numbers. They are named after the mathematicians Marin Mersenne (1588–1648) and Pierre de Fermat (1601–1665), respectively.
Mersenne primes can be written in the form of 2 n – 1, Where n is a prime number. This can be used in the proof of Fermat’s little theorem. In particular, the Mersenne prime is the largest number that is one less than a power of two. There are only 47 known Mersenne primes.
Fermat’s primes are those that can be written as F n = 2^k + 1 for some k where 1 ≤ k ≤ n. These are also called “regular primes” or “sporadic primes”. There is no known formula for testing if a given number is a Fermat prime, but there is an efficient algorithm to test if it is composite (i.e., not prime).
Difference Between Prime Number and Composite Number
Prime Numbers: Prime numbers are the natural numbers greater than one that is not divisible by any integer other than 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.
Composite Numbers: Composite Numbers are natural numbers greater than one that can be divided exactly by two integers other than 1 and itself. The first few composite numbers are 4, 6, 8, 9, 10, 12, and so on.
Conclusion
Prime numbers are important in many areas of the natural sciences, including physics and chemistry. In particular, they can be used to construct periodic patterns in atomic spectra and crystal structures. Prime numbers are also essential for public-key cryptography which is a fundamental tool for internet security. The RSA cryptosystem used by Facebook, Google, and many other companies relies on the fact that it’s very difficult to factor large composite numbers into their primes; this is why we can trust that our encrypted messages won’t be intercepted by an attacker trying to break into our accounts.”