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What is a Prime Number

What is a Prime Number

The question of what prime numbers are, takes us deep into mathematics territory. In this article, we explain what's special about these numbers.
Omkar Phatak
One of the most fundamental concepts in mathematics which you must grasp, is that of a prime number. Mathematics and number theory in particular is devoted to their study, as one of the most exciting of research topics.
What are They?
A prime number is any natural number that is fully divisible only by itself and the number 1. In other words, it's a number which cannot be factorized into other numbers. For example, 2 is only divisible by 2 itself and 1. 2 can only be factorized into 2 and 1 (i.e. 2 = 2 x 1). So two is in fact the smallest one of them.
You may ask, why 1 isn't a prime number? Though it fits the definition, by convention it is not considered to be one. It was proved by Euclid, centuries ago, that these numbers are infinite. A composite number is fully divisible by natural numbers, other than itself and 1.
To determine whether a number is a prime or composite, you need to factorize it. If it turns out that it has factors other than itself and 1, it is the latter. If not, then it belongs to the former category.
For example, let us see if 9 belongs to the category. On factorization, we see that 9 = 3 x 3 x 1. So 9 definitely isn't a prime number. What about 13? If you try factorization of 13, you discover that 13 = 13 x 1. That means, 13 belongs to the category.
Prime Number Theorem
Known as the 'Fundamental Theorem of Arithmetic', it states that any number (which is greater than 1), can be factorized into a product of prime numbers and this product is unique. So every composite number, is a unique product of these numbers and their powers. For example, 15 = 1 x 3 x 5, which is a unique product. Remember this theorem as it is one of the most important theorems in all of arithmetic.
Chart
So how many prime numbers are there? Mathematicians have pondered about this question for years and proved that they are indeed infinite. Here is a chart showing all those which are lesser than 1000.
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
877 881 883 887 907 911 919 920 937 941
947 953 967 971 977 983 991 997

Prime numbers can be looked at as building blocks of composite numbers. There are many interesting theorems and properties related to them, that you should know about.