## Twin Primes

A prime number (such as 2 or 3 or 5) is a natural number (positive whole number) which is only divisible by itself and one (and no other natural number). The other natural numbers (greater than one), are called composite numbers, and are the products of prime numbers. One is neither prime nor composite. See The Infinitude Of Primes and Sieve Of Eratosthenes and Primes and 6. Here are the first few primes:

```   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```

Twin primes are pairs of primes which differ by two (such as 3 and 5). Here I have made the first of a pair bold:

```   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```

As the numbers get larger, the primes become rarer. Naturally, the same thing happens to the twin primes. It is well known that there are infinitely many primes (The Infinitude Of Primes). It is not known whether there are infinitely many twin primes or not.

3,5 and 5,7 are two pairs of twin primes that share a number (5). Are there any other pairs of twin primes like that (n, n+2, n+4)? The answer is that there are no other triplet primes like that. In Primes and 6, we see that after the first few primes, every twin prime pair has a multiple of 6 immediately between them (6n-1, 6n, and 6n+1). Two greater than the second prime of a twin prime pair would be 6n+3 which is a multiple of 3, and not prime.