__Chronography of Prime Numbers__

__Page last modified 22
January 2023__

__2018__, The largest **prime number**
so far known was calculated by Patrick Laroche. It had **24,862,048 digits**.

__1989__, A **prime number** with **65,087**
digits was calculated at the Amdahl Corporation, California, USA.

__1985__, The number composed of 1,031 ones in a row
was found to be** prime**.

__1985__, The largest-then-known **prime number**,(2
to the power 216,065) minus 1, with 66,050 digits,was discovered.

__1896__, **Jacques Hadamard** proved that, for large values
of a, the number of **primes** less than a approximates to a / log a.

__1859__, **Bernhardt Reimann** (1826-66) introduced the **Reimann Hypothesis**, which predicted the
frequency of **prime numbers** amongst all integers.

__1648__, Death of French monk **Marin Mersenne** (born 1588), who first
identified what are now know as **Mersenne Prime Numbers**; those whose formula is 2^{n}
� 1.

__1588__, Italian mathematician **Pietro Cataldi** discovedred the **largest known prime number, 524,287**. It
remained the largest-known prime for almost two centuries.

__323 BCE__, **Euclid**
published his work �*Elements�*, the
standard text on **geometry**. He proved
that there must be infinitely many **prime numbers**.

**Overview of prime
number frequencies** (1^{st}
50 million primes supplied by **Chris K. Caldwell**, caldwell@utm.edu

Mathematics and Statistics, University Tennessee at Martin)

N th prime |
Value |
Increase from previous row |
Global frequency |
Frequency from previous row |

10,000 th |
104,729 |
104,729 |
10.4729 |
10.4729 |

1,000,000 th |
15,485,683 |
15,380,954 |
15.48568 |
15.38095 |

2,000,000 th |
32,452,843 |
16,967,160 |
16.22642 |
16.96716 |

3,000,000 th |
49,979,687 |
17,526,844 |
16.6599 |
17.52684 |

4,000,000 th |
67,867,967 |
17,888,280 |
16.96699 |
17.88828 |

5,000,000 th |
86,028,121 |
18,160,154 |
17.20562 |
18.16015 |

6,000,000 th |
104,395,301 |
18,367,180 |
17.39922 |
18.36718 |

7,000,000 th |
122,949,823 |
18,554,522 |
17.56426 |
18.55452 |

8,000,000 th |
141,650,939 |
18,701,116 |
17.70637 |
18.70112 |

9,000,000 th |
160,481,183 |
18,830,244 |
17.83124 |
18.83024 |

10,000,000 th |
179,424,673 |
18,943,490 |
17.94247 |
18.94349 |

11,000,000 th |
198,491,317 |
19,066,644 |
18.04467 |
19.06664 |

12,000,000 th |
217,645,177 |
19,153,860 |
18.1371 |
19.15386 |

13,000,000 th |
236,887,691 |
19,242,514 |
18.22213 |
19.24251 |

14,000,000 th |
256,203,161 |
19,315,470 |
18.30023 |
19.31547 |

15,000,000 th |
275,604,541 |
19,401,380 |
18.37364 |
19.40138 |

16,000,000 th |
295,075,147 |
19,470,606 |
18.4422 |
19.47061 |

17,000,000 th |
314,606,869 |
19,531,722 |
18.50629 |
19.53172 |

18,000,000 th |
334,214,459 |
19,607,590 |
18.56747 |
19.60759 |

19,000,000 th |
353,868,013 |
19,653,554 |
18.62463 |
19.65355 |

20,000,000 th |
373,587,883 |
19,719,870 |
18.67939 |
19.71987 |

21,000,000 th |
393,342,739 |
19,754,856 |
18.73061 |
19.75486 |

22,000,000 th |
413,158,511 |
19,815,772 |
18.77993 |
19.81577 |

23,000,000 th |
433,024,223 |
19,865,712 |
18.82714 |
19.86571 |

24,000,000 th |
452,930,459 |
19,906,236 |
18.8721 |
19.90624 |

25,000,000 th |
472,882,027 |
19,951,568 |
18.91528 |
19.95157 |

26,000,000 th |
492,876,847 |
19,994,820 |
18.9568 |
19.99482 |

27,000,000 th |
512,927,357 |
20,050,510 |
18.99731 |
20.05051 |

28,000,000 th |
533,000,389 |
20,073,032 |
19.03573 |
20.07303 |

29,000,000 th |
553,105,243 |
20,104,854 |
19.07259 |
20.10485 |

30,000,000 th |
573,259,391 |
20,154,148 |
19.10865 |
20.15415 |

31,000,000 th |
593,441,843 |
20,182,452 |
19.14329 |
20.18245 |

32,000,000 th |
613,651,349 |
20,209,506 |
19.1766 |
20.20951 |

33,000,000 th |
633,910,099 |
20,258,750 |
19.2094 |
20.25875 |

34,000,000 th |
654,188,383 |
20,278,284 |
19.24083 |
20.27828 |

35,000,000 th |
674,506,081 |
20,317,698 |
19.2716 |
20.3177 |

36,000,000 th |
694,847,533 |
20,341,452 |
19.30132 |
20.34145 |

37,000,000 th |
715,225,739 |
20,378,206 |
19.33043 |
20.37821 |

38,000,000 th |
735,632,791 |
20,407,052 |
19.35876 |
20.40705 |

39,000,000 th |
756,065,159 |
20,432,368 |
19.38629 |
20.43237 |

40,000 000 th |
776,531,401 |
20,466,242 |
19.41329 |
20.46624 |

41,000,000 th |
797,003,413 |
20,472,012 |
19.43911 |
20.47201 |

42,000,000 th |
817,504,243 |
20,500,830 |
19.46439 |
20.50083 |

43,000,000 th |
838,041,641 |
20,537,398 |
19.48934 |
20.5374 |

44,000,000 th |
858,599,503 |
20,557,862 |
19.51363 |
20.55786 |

45,000,000 th |
879,190,747 |
20,591,244 |
19.53757 |
20.59124 |

46,000,000 th |
899,809,343 |
20,618,596 |
19.56107 |
20.6186 |

47,000,000 th |
920,419,813 |
20,610,470 |
19.5834 |
20.61047 |

48,000,000 th |
941,083,981 |
20,664,168 |
19.60592 |
20.66417 |

49,000,000 th |
961,748,927 |
20,664,946 |
19.62753 |
20.66495 |

50,000,000 th |
982,451.653 |
20,702,726 |
19.64903 |
20.70273 |

**Click here for spreadsheet of more prime
number frequencies**

**Notes, 1) For each 1,000
integers the running tally of gaps between the primes is given (top row). The
only odd gap 1, between primes 2 and 3, is omitted). EXAMPLE, the first 3,000
integers contain 41 pairs of primes that are 10 integers apart.**

**2) Right hand column,
numbers in red, are the average number of integers per prime. EXAMPLE, for the
first 7,000 integers, the primes are average 7.7951 integers apart.**

**3) Numbers highlighted in
yellow are the proportion of gaps between primes having certain values, EXAMPLE
for the first 6,000 integers, there are 192 pairs of primes that are 5 digits
apart and this 192 comprises 0.2458 of the total number of prime pairs for
integers 0 � 5,999.**

**The graphic at foot of this
spreadsheet**** **shows the **evolving frequency of the gaps** between
the primes as one progresses up the integer range, e.g. the commonest gap
between primes is 6, but the frequency of this gap declines from 0.265 (26.5%
of total gaps) down to nearer 0.2013 (20.13%) of the total as one moves from
interger 0 to integer ** 105,997 **(10,106 th prime)

The 10-gap appears to be a sort of
�pivot� � gaps below this are becoming less common, but gaps larger than 10 are
becoming more common, as one progresses up the integer range.

__Two intriguing possibilities with prime numbers, one of which
must be true__**. Either,
One, there is some pattern to them, which may emerge at higher integer ranges
(that might mess up some security coding stuff), OR, Two, if there is no
pattern, then by colour coding the gaps an image of, literally, every
conceivable object in the universe will be depicted in 2-d by the resultant
graphics, so long as you go high enough up the integer range **

**Click here for some
graphics generated by prime numbers Each square is coloured according to
the size of the gap it represents between consecutive primes, e.g. 2 = dark
green, 4 = mid green, 6 = pale green, 8 = yellow etc. The coloured square
arrays are in row order of gaps between prime numbers, e.g. 1 ^{st} gap,
2^{nd} gap, 3^{rd} gap, 4^{th} gap��9^{th} gap
would be in the 3x3 array**

**1 ^{st} 2^{nd} 3^{rd}**

**4 ^{th} 5^{th} 6^{th}**

**7 ^{th} 8^{th} 9^{th}**

**In a 3x3 array.**

**Some other useful prime
number sites**

First 50 million primes, https://primes.utm.edu/lists/small/millions/

**And if you�re really keen**, here they are up to 1,000
billion, Yep, one trillion. That�s heavy��

http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php

https://www.numberempire.com/primenumbers.php