The Project Gutenberg eBook, Primes to One Trillion, Edited by Don Kostuch

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Title: Primes to One Trillion

Editor: Don Kostuch

Release Date: November 3, 2018 [eBook #58225]

Language: English

Character set encoding: ASCII

***START OF THE PROJECT GUTENBERG EBOOK PRIMES TO ONE TRILLION***

Primes to One Trillion

The Gutenberg text "The First 100,000 Prime Numbers", EBook #65, lists the primes up to 1,318,699. This somewhat more ambitious version lists the primes up to one trillion (1,000,000,000,000 or 1E12).

Introduction

I became interested in prime numbers after hearing about Goldbach's Conjecture,

"Every even integer greater than 2 can be expressed as the sum of two primes".

Verifying this requires a source of primes. Short lists (or programs to generate them) are widely available. Really long lists are scarce, except for
primos.mat.br.

To make these lists more accessible, I have reformatted them to a size easily manageable by ordinary text editors and viewers--about 55MB. The file names correspond to the range of primes the file contains:

00000000000_to_00100000000.txt

Zero to 100 Million

00100000000_to_00200000000.txt

100 Million to 200 Million

etc.

The leading zero digits cause the file names to collate in order of their content. Longer lists can be composed with the DOS copy command. Move the required prime txt files to a temporary directory and use:

copy *.txt longList.txt

Prime Text Files

This is a collection of 10,000 files, occupying about 486GB of disk space in their unzipped native txt format. Since adjacent primes have about 90% identical leading digits, the compressed (zip) versions total 61GB. Each zip file contains 100 txt files.

Do not use Windows Explorer to copy or move large numbers of files at a time. Use DOS copy or xcopy for large copies. I find Beyond Compare (Scooter Software) handy for keeping track of large numbers of files.

The following zip file names denotes the start of a range of primes in billions (1,000,000,000), and covers a range of 10,000,000,000. Each zip file contains 100 txt files, each covering the primes in a range of 100,000,000.

0000.zip    000G to  010G (0 to 1E10)
0010.zip    010G to  020G
0020.zip    020G to  030G
0030.zip    030G to  040G
0040.zip    040G to  050G
0050.zip    050G to  060G
0060.zip    060G to  070G
0070.zip    070G to  080G
0080.zip    080G to  090G
0090.zip    090G to  100G

0100.zip    100G to  110G
0110.zip    110G to  120G
0120.zip    120G to  130G
0130.zip    130G to  140G
0140.zip    140G to  150G
0150.zip    150G to  160G
0160.zip    160G to  170G
0170.zip    170G to  180G
0180.zip    180G to  190G
0190.zip    190G to  100G

0200.zip    200G to  210G
0210.zip    210G to  220G
0220.zip    220G to  230G
0230.zip    230G to  240G
0240.zip    240G to  250G
0250.zip    250G to  260G
0260.zip    260G to  270G
0270.zip    270G to  280G
0280.zip    280G to  290G
0290.zip    290G to  200G

0300.zip    300G to  310G
0310.zip    310G to  320G
0320.zip    320G to  330G
0330.zip    330G to  340G
0340.zip    340G to  350G
0350.zip    350G to  360G
0360.zip    360G to  370G
0370.zip    370G to  380G
0380.zip    380G to  390G
0390.zip    390G to  300G

0400.zip    400G to  410G
0410.zip    410G to  420G
0420.zip    420G to  430G
0430.zip    430G to  440G
0440.zip    440G to  450G
0450.zip    450G to  460G
0460.zip    460G to  470G
0470.zip    470G to  480G
0480.zip    480G to  490G
0490.zip    490G to  400G

0500.zip    500G to  510G
0510.zip    510G to  520G
0520.zip    520G to  530G
0530.zip    530G to  540G
0540.zip    540G to  550G
0550.zip    550G to  560G
0560.zip    560G to  570G
0570.zip    570G to  580G
0580.zip    580G to  590G
0590.zip    590G to  500G

0600.zip    600G to  610G
0610.zip    610G to  620G
0620.zip    620G to  630G
0630.zip    630G to  640G
0640.zip    640G to  650G
0650.zip    650G to  660G
0660.zip    660G to  670G
0670.zip    670G to  680G
0680.zip    680G to  690G
0690.zip    690G to  600G

0700.zip    700G to  710G
0710.zip    710G to  720G
0720.zip    720G to  730G
0730.zip    730G to  740G
0740.zip    740G to  750G
0750.zip    750G to  760G
0760.zip    760G to  770G
0770.zip    770G to  780G
0780.zip    780G to  790G
0790.zip    790G to  700G

0800.zip    800G to  810G
0810.zip    810G to  820G
0820.zip    820G to  830G
0830.zip    830G to  840G
0840.zip    840G to  850G
0850.zip    850G to  860G
0860.zip    860G to  870G
0870.zip    870G to  880G
0880.zip    880G to  890G
0890.zip    890G to  800G

0900.zip    900G to  910G
0910.zip    910G to  920G
0920.zip    920G to  930G
0930.zip    930G to  940G
0940.zip    940G to  950G
0950.zip    950G to  960G
0960.zip    960G to  970G
0970.zip    970G to  980G
0980.zip    980G to  990G
0990.zip    990G to 1000G

PrimeC File Format and
Miscellaneous C++ Programs

While working with primes, I developed the primec format, a file or array representation for primes that is roughly the same size of the compressed (zipped) txt representation, and supports fast access, both sequential and direct. The exact location of the primality specification of any number in the file (or memory array) is computed with a few instructions and no search.

If you wish to examine and experiment with the C++ programs used to reformat these prime lists and test the Goldbach Conjecture, download the "programs.zip" package. It contains Generating and Analyzing Prime Numbers, a description of the content and use of these files, including the primec file format.

Primec Format

The primec format exploits the fact that all primes greater than 5 end in the decimal digits 1, 3, 7, or 9. Thus, the primality of 20 successive numbers can be specified in one 8 bit byte. The file begins with the complete binary representation of:

The beginning of the sequence
The end of the sequence
A check sum of all data bytes
(All three are 8 bytes for this implementation).

The first and last values are a multiple of twenty, thus are never primes. There is no overlap of primes between successive files that use the same number for the upper boundary of the first file, and the lower boundary of the second file.

The primality of any number in the range of the file is determined as follows:

If the number ends in 0, 2, 4, 5, 6, or 8, it is not prime.

Otherwise, the location of the specifying byte is at offset:

( value - start ) / 20

Within that byte, the primality of the value is specified by the bit as shown in the following table.

The only tedious programming tasks were:

Special case code for values less than 20, which include 2 and 5, and exclude 1 and 9. All larger values follow the same simple pattern.

The increment and decrement operators for the corresponding iterators must search forward (or backward) for the next true bit, specifying the next prime number.

This table shows the layout and content for a file containing 20 to 60. The first 24 bytes (start value, end value, check sum) are not shown.

Byte

0

1

Bit

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Value

21

23

27

29

31

33

37

39

41

43

47

49

51

53

57

59

Prime

F

T

F

T

T

F

T

F

T

T

T

F

F

T

F

T

Hex

5A

E5

As primes become larger, the density of primes becomes smaller as 1/ln(n). Thus the density of true bits also falls off. The number of digits (binary or decimal) to represent the primes grows as ln(n). Thus, a sequence of primes represented as primec is always competitive in size with the corresponding sequence in ASCI text or binary, besides providing fast direct access by value:

bool isPrime(value).

The results for the sequence of the largest 64 bit primes (18446744073707000000 to 18446744073709551558) is:

Format

Size
(KB)

64 Bit Binary

445

Txt

1150

Zip Txt

114

PrimeC

125

Programs

Among the programs in the "program.zip" package are:

BuildTxtPrime

Create file of primes, txt format

TxtToPrimeC

Convert txt to primec format.

Goldbach

Verify Goldbach's conjecture
for zero to 1E12

Among the more than 15 classes and utilities are:

PrimeGenerator

Create prime numbers in a given range.

Directory

A vector of strings containing the names of files in a file directory.

Progress

A class to manage the periodic reporting of program activity.

PrimeCVector

Abstract class providing the algorithms to access primec data.

PrimeCFileWriter

Create a primec file.

PrimeCFileReader

Read a primec file.

I hope you find them useful.

If you have any questions, observations or bug reports concerning the C++ programming or the content of the prime files, send an email (after changing "at" to "@".

primes1e12 at earthlink.net

I embarked on this project as a programming challenge. I am not a mathematician. I have no deep insight into prime number theory. Please confine messages to programming issues. Here are some references:

Prime Numbers
Wikipedia: List of Prime Numbers (with numerous references)
The Math Forum
The Prime Pages

The program files are also posted on
http://home.earthlink.net/~primes1E12.
Corrections and additions will be posted there as they occur.

Don Kostuch
October, 2018.

***END OF THE PROJECT GUTENBERG EBOOK PRIMES TO ONE TRILLION***

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