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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 composite is 38, i.e. all of 380, 381, 382, ..., 38E are composite. (numbers k such that all of 10k, 10k + 1, 10k + 2, ..., 10k + E are composite are 38, 5X, 60, 62, 89, 93, 94, E0, E5, E8, ...)

The maximal gaps of primes less than 1000 is (927, 955), the gap (the difference of these two numbers) is 2X (i.e. there are 29 consecutive composites between 927 and 955).

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 section are also in duodecimal, i.e. 20% means 0.2 or 20/100 = 1/6, 36% means 0.36 or 36/100 = 7/20, 58.7% means 0.587 or 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 Template:Oeis, 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 Template:Oeis, 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 dp 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 0 and 100, 23 primes between 100 and 200, 1X primes between 200 and 300, 1X primes between 300 and 400, 1E primes between 400 and 500, 1X primes between 500 and 600, 16 primes between 600 and 700, 1X primes between 700 and 800, 18 primes between 800 and 900, 16 primes between 900 and X00, 1X primes between X00 and E00, 17 primes between E00 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

$ \frac{10^n}{n \cdot ln(10)} $ primes less than 10n (i.e. with at most n digits), and ln(10) = 2.599E035E8169...

N total numbers of primes less than N numbers of primes end with 1 less than N numbers of primes end with 5 less than N numbers of primes end with 7 less than N numbers of primes end with E less than 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

The twin prime pairs below 100 are (3, 5), (5, 7), (E, 11), (15, 17), (25, 27), (35, 37), (4E, 51), (5E, 61), (85, 87), (8E, 91), (E5, E7), and there are only 47 twin prime pairs below 1000. Except (3, 5), all twin prime pairs end with (5, 7) or (E, 1), and it is conjectured that the density of them are the same.

Classes of prime numbers Edit

  • By formula:
    • Fermat primes: of the form 22n + 1
      • Generalized Fermat primes base b: of the form b2n + 1
    • Mersenne primes: of the form 2p − 1 with prime p
      • Generalized Mersenne primes (Generalized repunit primes) base b: of the form (bp − 1)/(b − 1)
    • Double Mersenne primes: of the form 22p − 1 − 1 with prime p
    • Wagstaff primes: of the form (2p + 1)/3 with odd prime p
      • Generalized Wagstaff primes base b: of the form (bp + 1)/(b + 1) with odd prime p
    • Factorial primes: of the form n! ± 1
    • Primorial primes: of the form p# ± 1 with prime p
    • Proth primes: of the form k×2n + 1 with odd k and 2n > k
      • Generalized Proth primes base b: of the form k×bn + 1 with k not divisible by b and bn > k
    • Riesel primes: of the form k×2n − 1 with odd k and 2n > k
      • Generalized Riesel primes base b: of the form k×bn − 1 with k not divisible by b and bn > k
    • Pythagorean primes: of the form 4n + 1 (i.e. end with 1 or 5)
    • Pierpont primes: of the form 2m×3n + 1
    • Cullen primes: of the form n×2n + 1
      • Generalized Cullen primes base b: of the form n×bn + 1 with nb−1
    • Woodall primes: of the form n×2n − 1
      • Generalized Woodall primes base b: of the form n×bn − 1 with nb−1
  • By size: