Since Ancient Greece, prime numbers have been a topic of interest. Prime numbers are important since they are the building blocks of whole numbers, and their odd mathematical properties make them perfect for technological use. Euclid first suggested and established the notion of prime numbers, which states that there exists an endless list of primes.
A brilliant scientist named Eratosthenes, who lived a few centuries after Euclid, invented a sophisticated prime number theorem for determining all the prime numbers up to a specific positive integer. This method is known as the Eratosthenes Sieve. Fast forward to a couple of thousand years later and here we are! With all that in mind, let us delve into prime numbers!
What is a prime number?
A prime number is a positive integer that has no positive integer divisors other than itself and 1. This means that a prime number is a positive integer that has one positive divisor other than 1. It is therefore a number that cannot be factored by other digits.
Consider the positive integer divisors of the number 7, which are 1 and 7. The integer is a prime number in this case. However, if we consider the number 8, whose factors are 1, 2, 4, and 8, it isn’t a prime number. When we look at the number 1, we can see that it only has one positive integer divisor. As a result, it isn’t a prime number because primes must have two elements.
You should also note that in number theory, the twin prime conjecture asserts that there are an unlimited number of twin primes or pairs of primes that differ by two. Examples of this include:
- 3 and 5
- 5 and 7
- 11 and 13
- 17 and 19
How are prime numbers used in the real world?
Prime numbers are used in the real world in a variety of ways:
- Cicadas use them to track their life cycles;
- Current screens use them to figure out pixel color intensities;
- Manufacturers use them to remove harmonics from their products;
- Cybersecurity uses prime numbers because they are irreducible.
Whatever your emotions are towards prime numbers, they are used in our everyday lives and they are an important part of our civilization because they are inextricably linked to the fabric of the cosmos.
List of prime numbers
Between the integers 1 and 100, there are exactly 25 prime numbers. The following is a complete list of prime numbers from 1 to 100:
Prime numbers between 1 and 20
- 2, 3, 5, 7, 11, 13, 17, 19
Prime numbers between 21 and 40
- 23, 29, 31, 37
Prime numbers between 41 and 60
- 41, 43, 47, 53, 59
Prime numbers between 61 and 100
- 61, 67, 71, 73, 79, 83, 89, 97
How to find common multiples of a number
Properties of prime numbers
- Prime numbers are whole numbers greater than 1.
- They have two factors, which are 1 and the number itself.
- The only even prime number is 2.
- All numbers can be expressed as the product of prime numbers.
- Any prime number pairs are co-prime to each other.
Prime numbers and co-prime numbers
It’s important to distinguish between prime numbers and co-prime numbers. Co-prime numbers are pairs of numbers that do not possess any common factor other than 1 among each other. Co-prime numbers do not have to be prime numbers to be co-prime, but any two prime numbers are co-prime to each other.
The following are examples of co-prime numbers:
- 5 and 9 are co-prime numbers.
- 6 and 11 are co-prime numbers.
- 18 and 35 are co-prime numbers.
Easy way to find prime numbers
There are several methods for locating prime numbers. Let’s have a look at two different approaches.
Method #1
In the formula ‘n2 + n + 41,’ replace n with a whole number. All prime integers greater than 40 can be found using this formula. Let’s check by substituting a few whole numbers.
- 0x2 + 0 + 41 = 0 + 41 = 41; where n=0
- 1×2 + 1 + 41 = 2 + 41 = 43; where n=1
- 2×2 + 2 + 41 = 6 + 41 = 47; where n=2
If you keep going in this direction, you’ll be able to calculate all prime numbers bigger than 40.
Method #2
Except for 2 and 3, any prime number can be written as ‘6n + 1’ or ‘6n – 1’. So, if you have a number that isn’t 2 or 3, try to express it in the form of 6n + 1 or 6n – 1 to see if it’s prime or not.
- 1 – 6(1) = 5
- 6(1) + 1 = 7
- 1 – 6(2) = 11
- 13 = 6(2) + 1
We now know that 5, 7, 11, and 13 are prime numbers.
List of odd prime numbers
| 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 |
This is a difficult question in part because, except for 2, all prime numbers are odd all the way to infinity. Apart from that, all other prime numbers are odd. The number 2 is also the smallest prime number.
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Largest prime number
The highest known prime number (as of March 2022) is 282,589,933 -1, which when expressed in base 10 has 24,862,048 digits. The first 120 digits of its value are shown below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 … (24,861,808 digits remain)
Anyone who downloads and executes the Great Internet Mersenne Prime Search (GIMPS) software and discovers a new Mersenne prime with fewer than 100 million digits will get a 3,000 USD research discovery reward.
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