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prime numbers
By definition a number  $p$  is prime if  $p>1$  and its only divisors are 1 and  $p$  itself.

There are infinite prime numbers.

Stanislaw Ulam in 1963 observed that drawing the integers along a spiral (as depicted below) the prime numbers often seem to align along some diagonals. Essentially this depends on the fact that certain quadratic polynomials produce (at least in a limited range) more primes than others.

prime spiral  prime spiral
A spiral with side 256:
prime spiral
This construction can be applied to other families of numbers, like abundant, admirable, alternating, apocalyptic, arithmetic, binomials, congruent, deficient, balanced, economical, equidigital, evil, hoax, inconsummate, junction, lucky, nialpdrome, odious, plaindrome, practical, pseudoperfect, self, super-d, tau, triangular, Ulam, wasteful and Zumkeller numbers.

The prime numbers up to 100 are 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. more terms

Prime numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 13 101 + 9999993463 aban 11 13 17 + 999999000961 alt.fact. 19 101 619 + 3301819 alternating 23 29 41 + 989898989 amenable 13 17 29 + 999999937 apocalyptic 157 251 443 + 29989 arithmetic 11 13 17 + 9999991 balanced p. 53 157 173 + 9999997543 Bell 877 27644437 bemirp 1061 1091 1601 + 199999069901 c.decagonal 11 31 61 + 999995100781 c.heptagonal 43 71 197 + 999977610703 c.pentagonal 31 181 331 + 999977762881 c.square 13 41 61 + 999982234201 c.triangular 19 31 109 + 999980205949 Carol 47 223 3967 + 274876858367 Chen 11 13 17 + 99999971 congruent 13 23 29 + 9999991 constructible 17 257 65537 Cunningham 17 31 37 + 999920001601 Curzon 29 41 53 + 199999853 cyclic 11 13 17 + 9999991 d-powerful 43 89 283 + 9993623 de Polignac 127 149 251 + 99999839 deficient 11 13 17 + 9999991 dig.balanced 11 19 37 + 199999777 economical 11 13 17 + 19999999 emirp 13 17 31 + 199999931 equidigital 11 13 17 + 19999999 esthetic 23 43 67 + 898989898987 Eulerian 11 1013 15619 + 67108837 evil 17 23 29 + 999999937 fibodiv 19 47 61 + 8613332801 Fibonacci 13 89 233 + 2971215073 Friedman 127 347 2503 + 995347 Gilda 29 683 997 + 271953103 good prime 11 17 29 + 199968539 happy 13 19 23 + 9999991 hex 19 37 61 + 999962694781 Hogben 13 31 43 + 999999000001 Honaker 131 263 457 + 999843421 house 271 hungry 17 2003 iban 11 17 23 + 777743 iccanobiF 13 4139 9291169663 idoneal 13 37 inconsummate 173 383 431 + 999953 Jacobsthal 11 43 683 + 715827883 junction 101 103 107 + 99999259 katadrome 31 41 43 + 98765431 Kynea 23 79 1087 + 68720001023 Leyland 17 593 32993 + 8589935681 lonely 23 53 211 + 491856414677 Lucas 11 29 47 + 119218851371 lucky 13 31 37 + 9999823 m-pointer 23 61 1123 + 711611111111 magnanimous 11 23 29 + 608844043 metadrome 13 17 19 + 23456789 modest 13 19 23 + 1999633333 Motzkin 127 15511 nialpdrome 11 31 41 + 999999999961 nude 11 oban 11 13 17 + 997 odious 11 13 19 + 999999893 Ormiston 1913 1931 18379 + 1999996897 palindromic 11 101 131 + 99999199999 palprime 11 101 131 + 99999199999 pancake 11 29 37 + 999997083367 panconsummate 11 23 31 + 1291 pandigital 11 19 partition 11 101 17977 + 228204732751 pernicious 11 13 17 + 9999991 Perrin 17 29 277 + 1442968193 persistent 10123457689 10123465789 10123485679 + 98876532401 Pierpont 13 17 19 + 880602513409 plaindrome 11 13 17 + 788888888899 primeval 13 37 107 + 100123456789 Proth 13 17 41 + 999977648129 repdigit 11 repfigit 19 47 61 + 1084051 repunit 13 31 43 + 999999000001 self 31 53 97 + 999999937 self-describing 10153331 10173133 10233221 + 622929272627 sliding 11 29 101 641 star 13 37 73 + 999991228573 straight-line 4567 76543 23456789 strobogrammatic 11 101 181 + 699908806669 strong prime 11 17 29 + 99999959 super-d 19 31 107 + 9999931 tetranacci 29 401 773 5350220959 tribonacci 13 149 trimorphic 251 499 751 + 574218751 truncatable prime 13 17 23 + 999631686353 twin 11 13 17 + 999999193 uban 11 13 17 + 1000000000039 Ulam 11 13 47 + 9999877 undulating 101 131 151 + 91919191919 upside-down 19 37 73 + 99998521111 weak prime 13 19 23 + 99999971 weakly prime 294001 505447 584141 + 99999446179 Wieferich 1093 3511 Woodall 17 23 191 + 824633720831 Zuckerman 11 zygodrome 11 11177 11777 + 999999988811