## FANDOM

50 Pages

A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Natural numbers greater than 1 that are not prime are called composite.

The first 1X5 prime numbers (all the prime numbers less than 1000) are:

2, 3, 5, 7, E, 11, 15, 17, 1E, 25, 27, 31, 35, 37, 3E, 45, 4E, 51, 57, 5E, 61, 67, 6E, 75, 81, 85, 87, 8E, 91, 95, X7, XE, E5, E7, 105, 107, 111, 117, 11E, 125, 12E, 131, 13E, 141, 145, 147, 157, 167, 16E, 171, 175, 17E, 181, 18E, 195, 19E, 1X5, 1X7, 1E1, 1E5, 1E7, 205, 217, 21E, 221, 225, 237, 241, 24E, 251, 255, 25E, 267, 271, 277, 27E, 285, 291, 295, 2X1, 2XE, 2E1, 2EE, 301, 307, 30E, 315, 321, 325, 327, 32E, 33E, 347, 34E, 357, 35E, 365, 375, 377, 391, 397, 3X5, 3XE, 3E5, 3E7, 401, 40E, 415, 41E, 421, 427, 431, 435, 437, 447, 455, 457, 45E, 465, 46E, 471, 481, 485, 48E, 497, 4X5, 4E1, 4EE, 507, 511, 517, 51E, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58E, 591, 59E, 5E1, 5E5, 5E7, 5EE, 611, 615, 617, 61E, 637, 63E, 647, 655, 661, 665, 66E, 675, 687, 68E, 695, 69E, 6X7, 6E1, 701, 705, 70E, 711, 71E, 721, 727, 735, 737, 745, 747, 751, 767, 76E, 771, 775, 77E, 785, 791, 797, 7X1, 7EE, 801, 80E, 817, 825, 82E, 835, 841, 851, 855, 85E, 865, 867, 871, 881, 88E, 8X5, 8X7, 8XE, 8E5, 8E7, 901, 905, 907, 90E, 91E, 921, 927, 955, 95E, 965, 971, 987, 995, 9X7, 9XE, 9E1, 9E5, 9EE, X07, X0E, X11, X17, X27, X35, X37, X3E, X41, X45, X4E, X5E, X6E, X77, X87, X91, X95, X9E, XX7, XXE, XE7, XEE, E11, E15, E1E, E21, E25, E2E, E31, E37, E45, E61, E67, E6E, E71, E91, E95, E97, EX5, EE5, EE7

Except 2 and 3, all primes end with 1, 5, 7 or E. The first k such that all of 10k, 10k + 1, 10k + 2, ..., 10k + E are all composite is 38, i.e. all of 380, 381, 382, ..., 38E are composite.

The density of primes end with 1 is a relatively low (< 1/4), but the density of primes end with 5, 7 and E are nearly equal (all are a little more than 1/4). (i.e. for a given natural number N, the number of primes end with 1 less than N is usually smaller than the number of primes end with 5 (or 7, or E) less than N) e.g. For all 1426 primes < 10000, there are 3E8 primes (2E.3%) end with 1, 410 primes (30.3%) end with 5, 412 primes (30.5%) end with 7, 406 primes (2E.E%) end with E. It is conjectured that for every natural number N ≥ 10, the number of primes end with 1 less than N is smaller than the number of primes end with 5 (or 7, or E) less than N. (Note: the percentage in this sequence are also in duodecimal, i.e. 20% means 0.2 or {\displaystyle {\frac {20}{100}}={\frac {1}{6}}}, 36% means 0.36 or {\displaystyle {\frac {36}{100}}={\frac {7}{20}}}, 58.7% means 0.587 or {\displaystyle {\frac {587}{1000}}})

13665 is the smallest prime p such that the number of primes end with 1 or 5 ≤ p is more than the number of primes end with 3, 7 or E ≤ p (see OEIS: A007350, of course, 3 is the only prime ends with 3). Besides, 9X03693X831 is the smallest prime p such that the number of primes end with 1 or 7 ≤ p is more than the number of primes end with 2, 5 or E ≤ p (see OEIS: A007352, of course, 2 is the only prime ends with 2). Question: What is the smallest prime p such that the number of primes end with 1 ≤ p is more than the number of primes end with d ≤ p for at least one of d = 5, 7 or E?

All squares of primes (except 2 and 3) end with 1.

There are 2X primes between 1 and 100, 23 primes between 101 and 200, 1X primes between 201 and 300, 1X primes between 301 and 400, 1E primes between 401 and 500, 1X primes between 501 and 600, 16 primes between 601 and 700, 1X primes between 701 and 800, 18 primes between 801 and 900, 16 primes between 901 and X00, 1X primes between X01 and E00, 17 primes between E01 and 1000.

There are about N/ln(N) primes less than N, where ln is the natural logarithm, i.e. the logarithm with base e = 2.875236069821... (see prime number theorem), thus there are about {\displaystyle {\frac {10^{n}}{n\cdot ln10}}} primes less than 10n (i.e. with at most n digits), and ln(10) = 2.599E035E8169...

 N total numbers of primes ≤ N numbers of primes end with 1 ≤ N numbers of primes end with 5 ≤ N numbers of primes end with 7 ≤ N numbers of primes end with E ≤ N 10 5 0 1 1 1 40 13 2 4 4 3 100 2X 6 9 9 8 400 89 1X 23 23 23 1000 1X5 51 59 59 58 4000 621 157 16X 170 166 10000 1426 3E8 410 412 406 40000 4833 11X4 121E 1219 1211 100000 10852 31X4 3225 3225 321X 400000 3928E E333 E377 E3E9 E3X2 1000000 X4E20 27204 27295 2730X 27333

In the following table, numbers shaded in cyan are primes.

 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24 25 26 27 28 29 2X 2E 30 31 32 33 34 35 36 37 38 39 3X 3E 40 41 42 43 44 45 46 47 48 49 4X 4E 50 51 52 53 54 55 56 57 58 59 5X 5E 60 61 62 63 64 65 66 67 68 69 6X 6E 70 71 72 73 74 75 76 77 78 79 7X 7E 80 81 82 83 84 85 86 87 88 89 8X 8E 90 91 92 93 94 95 96 97 98 99 9X 9E X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 XX XE E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 EX EE 100

Although the prime ends with 1 are relatively low ...

Smallest prime obtained as a multiple of n followed by a 1 is 11, 61, 31, 81, 51, 61, 241, 81, 91, 181, 291, 301, 111, 241, 131, 141, 2X1, 301, 171, 181, 531, 921, 1E1, 401, ... (it is equivalent to the smallest prime of the form 20kn+1, where k is integer, such prime always exist by the Green-Tao theorem)

Smallest prime obtained as a power of n (>= 1st power) followed by a 1 is 11, 81, 31, 141, 51, 61, 14811, 81, 91, 841, X11, ... (the n=10 case is equivalent to the generalized Fermat prime in base 10, which has no known examples except 11, also, such prime does not exist for n = 1 mod 11 since the numbers are always divisible by 11, also not exist for n = EX mod EE since the numbers are always divisible by either E or 11, and if there are no generalized Fermat primes in base 10 except 11, then such prime does not exist for n that is a power of 10, however, such prime also not exists for n = 208, 3XE, 419, ... since these n have covering set)

The star primes are exactly the primes obtained as a triangular number followed by a 1: 11, 31, 61, 131, 241, 301, 391, 471, 661, ... (the star numbers are exactly the numbers obtained as a triangular number followed by a 1, the star numbers are exactly the centered dozagonal numbers (centered 10-gonal numbers))