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Seminar presented to the Department of Mathematics of the Federal University of Paraná, in 2006. Based on the paper "Prime Numbers without Mystery" first published in 2004 at Mathpreprints.com (no longer available).
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Seminars on Mathematics, UFPR, 2006Seminars on Mathematics, UFPR, 2006
Title: Title: Prime Numbers Without Mystery. Prime Numbers Without Mystery. (26/10/2006, Prof. (26/10/2006, Prof. Jânia Duha Jânia Duha ) )
Federal University of ParanaFederal University of Parana
Curitiba, PR, Brazil.Curitiba, PR, Brazil.
http://www.mat.ufpr.br/mestrado/palestras.htmlhttp://www.mat.ufpr.br/mestrado/palestras.html
Prime Numbers without Prime Numbers without MysteryMystery
Jania DuhaJania Duha20062006
IntroductionIntroduction From prehistoric man through ancient
Greecian times on into modern times, all through the world the elusive prime numbers have fascinated human minds.
Faced with the problem of determining when the next prime number will occur, many scientists have locked themselves in rooms, figuring, calculating, making logical moves or intuitive leaps.
What is a Prime Number?What is a Prime Number?
A selection of quotationsA selection of quotations
When reading these, take note of the When reading these, take note of the profusion of emotional/poetic/ecstatic and profusion of emotional/poetic/ecstatic and religiously-oriented language which is religiously-oriented language which is used throughout. The words used throughout. The words mystery, mystery, mysteriousmysterious and and secretssecrets appear numerous appear numerous times, but also times, but also sense of wonder, strange, sense of wonder, strange, stunning, astonishing, baffling, bafflement, stunning, astonishing, baffling, bafflement, devilment, surprise, endless surprises, devilment, surprise, endless surprises, exasperating, perplexingexasperating, perplexing … …
……great mystery, magic, alchemist, elixir, great mystery, magic, alchemist, elixir, aesthetic appeal, works of art, poetry, aesthetic appeal, works of art, poetry, arcane music, secret harmony, Nature's arcane music, secret harmony, Nature's gift, inexplicable secrets of creation, gem, gift, inexplicable secrets of creation, gem, gemstone, jewels, crown, heart, soul, gemstone, jewels, crown, heart, soul, cosmos, abyss(es), divine, Holy Grail, cosmos, abyss(es), divine, Holy Grail, Lucifer, DevilLucifer, Devil and and GodGod. .
"Prime numbers are the most basic objects in "Prime numbers are the most basic objects in mathematics. They also are among the most mathematics. They also are among the most mysteriousmysterious, for after centuries of study, the , for after centuries of study, the structure of the set of prime numbers is still not structure of the set of prime numbers is still not well understood. Describing the distribution of well understood. Describing the distribution of primes is at the heart of much mathematics..."primes is at the heart of much mathematics..."
A. Granville from AMS news release 5 December A. Granville from AMS news release 5 December 19971997
"Mathematicians have tried in vain to "Mathematicians have tried in vain to this day to discover some order in the this day to discover some order in the sequence of prime numbers, and we sequence of prime numbers, and we have reason to believe that it is a have reason to believe that it is a mysterymystery into which the mind will never into which the mind will never penetrate."penetrate."
Leonard Euler, in G. Simmons, Leonard Euler, in G. Simmons, Calculus GemsCalculus Gems, , McGraw-Hill, New York, 1992McGraw-Hill, New York, 1992
"Prime numbers have always "Prime numbers have always fascinated mathematicians. They fascinated mathematicians. They appear among the integers seemingly appear among the integers seemingly at random, and yet not quite: at random, and yet not quite: There There seems to be some order or pattern, just seems to be some order or pattern, just a little below the surface, just a little out a little below the surface, just a little out of reachof reach."."
Underwood Dudley, Underwood Dudley, Elementary Number TheoryElementary Number Theory (Freeman, 1978)(Freeman, 1978)
"The primes have tantalized "The primes have tantalized mathematicians since the Greeks, mathematicians since the Greeks, because they appear to be somewhat because they appear to be somewhat randomly distributed randomly distributed but not but not completely socompletely so."."
T. Gowers, T. Gowers, Mathematics: A Very Short IntroductionMathematics: A Very Short Introduction (Oxford Univ. Press, 2002), p.118(Oxford Univ. Press, 2002), p.118
"Who would have imagined that "Who would have imagined that something as straightforward as the something as straightforward as the natural numbers (1, 2, 3, 4,...) could natural numbers (1, 2, 3, 4,...) could give birth to anything so give birth to anything so bafflingbaffling as the as the prime numbers (2, 3 ,5, 7, 11, ...)?"prime numbers (2, 3 ,5, 7, 11, ...)?"
Ian Stewart, "Jumping Champions", Ian Stewart, "Jumping Champions", Scientific Scientific AmericanAmerican, December 2000, December 2000
"There are two facts about the distribution of prime "There are two facts about the distribution of prime numbers which I hope to convince you so numbers which I hope to convince you so overwhelmingly that they will be permanently overwhelmingly that they will be permanently engraved in your hearts.engraved in your hearts.
The first is that despite their simple definition and The first is that despite their simple definition and role as the building blocks of the natural numbers, role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the that of chance, and nobody can predict where the next one will sprout.next one will sprout.
The second fact is even more astonishing, for it The second fact is even more astonishing, for it states just the opposite: that the prime numbers states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these governing their behaviour, and that they obey these laws with almost military precision."laws with almost military precision."
Don Zagier, Bonn University inaugural lectureDon Zagier, Bonn University inaugural lecture
"Although the prime numbers are "Although the prime numbers are rigidly determined, they somehow feel rigidly determined, they somehow feel like experimental data."like experimental data."
T. Gowers, T. Gowers, Mathematics: A Very Short IntroductionMathematics: A Very Short Introduction (Oxford (Oxford Univ. Press, 2002), p.121Univ. Press, 2002), p.121
"It is evident that the primes are "It is evident that the primes are randomly distributed but, randomly distributed but, unfortunately, we don't know what unfortunately, we don't know what 'random' means.'''random' means.''
R. C. Vaughan (February 1990)R. C. Vaughan (February 1990)
"God may not play dice with the "God may not play dice with the universe, but something strange is universe, but something strange is going on with the prime numbers."going on with the prime numbers."
P. Erdös, referring to the famous quote of Einstein. From P. Erdös, referring to the famous quote of Einstein. From "Homage to an Itinerant Master" by D. Mackenzie "Homage to an Itinerant Master" by D. Mackenzie
((ScienceScience 275:759, 1997) 275:759, 1997)
"Given the millennia that people have "Given the millennia that people have contemplated prime numbers, our contemplated prime numbers, our continuing ignorance concerning the continuing ignorance concerning the primes is stultifying."primes is stultifying."
R. Crandall and C. Pomerance, from R. Crandall and C. Pomerance, from Prime Numbers: A Prime Numbers: A
Computational PerspectiveComputational Perspective (Springer-Verlag, 2001) (Springer-Verlag, 2001)
Eratosthenes (200 BC)Eratosthenes (200 BC)
Criou o mais famoso e eficiente algoritmo Criou o mais famoso e eficiente algoritmo para o calculo de numeros primospara o calculo de numeros primos
Sieve of EratosthenesSieve of Eratosthenes
Natural numbersNatural numbers
11 2 2 3 3 4 4 5 5 6 6 7 7 8… 8…
1 1+1 1+1+1 1+1+1+1 ….etc1 1+1 1+1+1 1+1+1+1 ….etc
The number “The number “11” is the main “” is the main “building building blockblock” for natural numbers. ” for natural numbers.
The numbers 2 and 3 are The numbers 2 and 3 are secondary building blockssecondary building blocks
2 4 6 8 10 122 4 6 8 10 12
1 2 3 4 5 6 7 8 9 10 11 12 …1 2 3 4 5 6 7 8 9 10 11 12 …
3 6 9 123 6 9 12
Something is missing…Something is missing…
11 2 3 2 3 4 4 55 6 6 77 8 9 10 8 9 10 1111 12 … 12 …
Note that some numbers are provided by the two and the three simultaneously. We will call these repeated numbers (6, 12, 18, 24, 30…)
as the "knots" of the sequence:
2 4 6 8 10 12 14 16 18 20 22 24
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25...
3 6 9 12 15 18 21 24
Knots “k”Knots “k” 66nn
Knot numbers “kn”Knot numbers “kn” 66nn 1 1
Composite knot numbers “kc”Composite knot numbers “kc” (6(6nn 1).(6 1).(6nn 1) 1)
n = 1, 2, 3, 4, 5, ....
Simple knot numbers “ks” areSimple knot numbers “ks” are
ks = kn - kcks = kn - kc
“ “ PRIME NUMBERS “PRIME NUMBERS “
Composite Knot NumbersComposite Knot Numbers
kn
kc
prime
0 50 100 150 200 250
PRIME NUMBERS PATTERN (Kc 5)
kn
kc
prime
0 50 100 150 200 250
PRIME NUMBERS PATTERN (Kc 5)
kn
kc
prime
0 50 100 150 200 250
PRIME NUMBERS PATTERN (Kc 7)
kn
kc
prime
0 50 100 150 200 250
PRIME NUMBERS PATTERN (Kc 7)
kn
kc
prime
49 259 469
kn
kc
prime
49 259 469
Number of primes for every thousand
y = 247.53x-0.0904 y = -8.1863Ln(x) + 182.66
0
40
80
120
160
200
0E+00 1E+05 2E+05 3E+05 4E+05 5E+05 6E+05
Natural numbers
Number of primes
Number of primes for every thousand
y = 247.53x-0.0904 y = -8.1863Ln(x) + 182.66
0
40
80
120
160
200
0E+00 2E+05 4E+05 6E+05 8E+05 1E+06
Natural numbers
Number of primes
Number of primes for every thousand
y = 247.53x-0.0904 y = -8.1863Ln(x) + 182.66
0
40
80
120
160
200
0E+00 2E+06 4E+06 6E+06 8E+06 1E+07
Natural numbers
Number of primes
Number of primes to a given number "n"
0
5000
10000
15000
20000
25000
30000
0 50000 100000 150000 200000 250000 300000
Natural numbers
Number of primes
104
105
106
107
108
109
106
107
108
109
1010
Natural numbers
Num
ber
of pri
mes
Number of primes to a given number "n"
106107 108 109 1010
104
105
106
107
108
109
Number of primes to a given number “n”
Blue line GaussPink marker Legendre
Pink marker Legendre
Blue line Gauss
(n) ~ n / lnn + n / (lnn)2 + 2n / (lnn)3 + ...
(n) ~ n / (ln n – 1,08366)
NotesNotes
Twin PrimesTwin Primesnn22 -1 -1alawys a prime between alawys a prime between nn and 2 and 2nnnn22 – – nn +41 for 0 +41 for 0 nn 40 40etc…etc…
The largest prime RECORDSThe largest prime RECORDS
Mersenne prime Mersenne prime MM2596495125964951 7 816 230 decimal digits 7 816 230 decimal digits
(February 2005)(February 2005)
Largest twin primes: Largest twin primes: 242206083 x 2242206083 x 23888038880 1 1 11713 digits (November, 1995)11713 digits (November, 1995)
Largest factorial prime (prime of the form n! Largest factorial prime (prime of the form n! 1) 1) 3610! - 13610! - 1 11277 digits (1993) 11277 digits (1993)
Largest primordial prime (prime of the form n# Largest primordial prime (prime of the form n# 1 where 1 where n# is product of all primes n# is product of all primes n) n) 24029# 24029# + 1 + 1 10387 digits (1993)10387 digits (1993)
ReferencesReferences
Duha, J. Duha, J. Prime numbers without mystery, Prime numbers without mystery, The The Mathematics Preprint Server, www.mathpreprints.com, Mathematics Preprint Server, www.mathpreprints.com, 2004.2004.
Duha, J. Duha, J. Prime numbers as potential pseudo-random Prime numbers as potential pseudo-random code for GPS signals, code for GPS signals, Boletim de Ciencias Geodesicas, Boletim de Ciencias Geodesicas, v.10, p.215-224, 2004.v.10, p.215-224, 2004.
Du Sautoy, M. Du Sautoy, M. The music of the primes The music of the primes, 2003., 2003.
"It will be millions of years before we'll have any understanding, and even then it won't be a complete
understanding, because we're up against the infinite."
P. Erdös (interview with P. Hoffman, Atlantic Monthly, Nov. 1987, p. 74)
