Is There a Formula that Is There a Formula that Generates Prime Generates Prime Numbers?Numbers?

A Sonoma State A Sonoma State M*A*T*H ColloquiumM*A*T*H Colloquium

Presentation byPresentation by

`tÜztÜxà bãxÇá`tÜztÜxà bãxÇáAssociate DeanAssociate Dean

College of Natural SciencesCollege of Natural Sciences

What What isis a prime number?a prime number?

A positive integer A positive integer pp is called a is called a prime numberprime number provided provided pp has has exactly two positive divisors.exactly two positive divisors.

pp11 == 2, 2, pp22 == 3, 3, pp33 == 5, 5, pp44 == 7, 7, pp55 == 11, ...11, ...

Is there a formula for primes?Is there a formula for primes?

Is there a function Is there a function f f such that such that ff( ( n n ) = ) = ppnn ,,the the nn--th prime, for allth prime, for all nn ??

Is there a function that produces only prime Is there a function that produces only prime numbers ?numbers ?

Are there polynomial functions that produce Are there polynomial functions that produce primes?primes?

Are there formulas for primes?Are there formulas for primes?

YES!YES!

Astonishing FormulasAstonishing Formulas

Amusing FormulasAmusing Formulas

Worthless FormulasWorthless Formulas

Some NotationSome Notation

Let Let nn be a positive integer.be a positive integer.Let Let nn! = 1! = 1⋅⋅22⋅⋅33⋅⋅ …… ⋅⋅((nn –– 1)1)⋅⋅nn..

Let Let ⎣⎣xx⎦⎦ represent the greatest integer that represent the greatest integer that does not exceed does not exceed xx..

A ToolA Tool

ππ((nn) = # primes not exceeding ) = # primes not exceeding nn

FindFind ππ((nn) for ) for nn = 1, 2, 3, 4, 5, 10, 20.= 1, 2, 3, 4, 5, 10, 20.

n 1 2 3 4 5 10 20

π(n) 0 1 2 2 3 4 8

A Worthless Formula for A Worthless Formula for ppnn

Find pFind p33..

∑=

⎥⎦

⎥⎢⎣

⎢

π++=

n

knn k

np2

1 )(11

Willans, 1964Willans, 1964

How Does It Work?How Does It Work?

WilsonWilson’’s Theorems Theorem (1770) (1770) nn is prime iffis prime iff [([(n n –– 11)! + 1] / )! + 1] / nn is an integer.is an integer.

Willans used WilsonWillans used Wilson’’s Theorem to count primes.s Theorem to count primes.

( ) ( )∑ ⎥

⎦

⎥⎢⎣

⎢ +−+−=

=

k

j j!j

cosk1

2 111 ππ

A Formula with A Formula with Encoded InformationEncoded Information

1952, 1952, SierpinskiSierpinski’’ss other constant other constant

A = 0.02030005000000070...A = 0.02030005000000070...

⇒∑=∞

=

−

110 2

k

kkpA

⎣ ⎦ ⎣ ⎦AAp nnnn 101010 222 11 −−

−=

Are There Functions that Are There Functions that Produce only Primes?Produce only Primes?

We want We want ff so that so that f f ( ( n n ) is always ) is always prime, but not necessarily prime, but not necessarily ppn n ..

We donWe don’’t want something like t want something like

f f ( ( n n ) = 17) = 17 for all for all n n ..

A Function that Produces A Function that Produces only Primesonly Primes

1951, Wright There exists a real 1951, Wright There exists a real numbernumberωω ≈≈ 1.9287800 so that the 1.9287800 so that the following function is prime for all following function is prime for all n .n .

( )⎥⎥⎦

⎥

⎢⎢⎣

⎢= 222 2ω

nf

Edw

ard

M. W

right

1906 20051906 2005

Another Function that Another Function that Produces only PrimesProduces only Primes

1947, Mills There exists a real 1947, Mills There exists a real numbernumber θθ ≈≈ 1.3064 so that the following 1.3064 so that the following function is prime for all function is prime for all n .n .

( ) ⎣ ⎦θ 3nng =

How about a Nice Function How about a Nice Function that Produces Lots of Primes?that Produces Lots of Primes?

1772, Euler1772, Euler’’s function s function f f ((n n ) = ) = n n 22 + + nn + 41+ 41

Theorem Theorem There is no nonThere is no non--constant constant polynomial in one variable with polynomial in one variable with integer coefficients which produces integer coefficients which produces only prime values for integer inputs.only prime values for integer inputs.

Polynomials that Polynomials that Generate PrimesGenerate Primes

1971, Matijasevic1971, MatijasevicThere exists a polynomial of degree 37 There exists a polynomial of degree 37 in 24 variables with integer coefficients in 24 variables with integer coefficients such that the set of prime numbers such that the set of prime numbers coincides with the set of positive values coincides with the set of positive values assumed by the polynomial as the assumed by the polynomial as the variables range in the set of nonvariables range in the set of non--negative integers.negative integers.

A Polynomial whose A Polynomial whose Positive Values are PrimePositive Values are Prime

1976 Jones, Sato, Wada, 1976 Jones, Sato, Wada, WiensWiens

found an explicit polynomial of found an explicit polynomial of degree 25 in 26 variablesdegree 25 in 26 variables

( ) [ ] ( )( )[ ]{[ ] ( ) ( )( )[ ]( )( )[ ] ( )[ ]

( )[ ] [ ]

( )( )( )( ) ( )[ ]( )[ ] [ ]

( ) ( )[ ]( ) ( )[ ]( ) ( )[ ] }22

22

22

22222

222222

222422

2222

2232

223 22

22

12

22221

22221

111

141

1116

11112

1121162

1212

pmpaptpaplz

xppaapspayq

mnnaanbnalp

ilkaimla

cuxdynauua

yvlnuayr

xyaoaee

fnkkezqpn

zhjhkggkqjhwzk

−−−+−+−

−−−−++−−+−

−−−−++−−+−

−−++−−+−−

−−++−−+−

−++−−+−−

−+−−−+++−

−+−++−−+++−

−+++++−−++−+

Is There a Formula that Is There a Formula that Generates Prime Numbers?Generates Prime Numbers?

Yes, there are many such Yes, there are many such formulas, but they all formulas, but they all seem to be worthless.seem to be worthless.

ReferencesReferences

Ribenboim, P., Are there Functions that Generate Ribenboim, P., Are there Functions that Generate Prime Numbers?, Prime Numbers?, College Math. JournalCollege Math. Journal 28:5 28:5 (1997) 352(1997) 352--359359

Dudley, U., History of a Formula for Primes, Dudley, U., History of a Formula for Primes, American Math. MonthlyAmerican Math. Monthly 76:1 (1969) 2376:1 (1969) 23--2828

Jones, Sata, Wada, Wiens, Diophantine Jones, Sata, Wada, Wiens, Diophantine Representation of the Set of Prime Numbers, Representation of the Set of Prime Numbers, American Math. MonthlyAmerican Math. Monthly 83:6 (1976) 44983:6 (1976) 449--464464

ReferencesReferences

Gupta, N., Gupta, N., Finding a Solution to the Finding a Solution to the Diophantine Representation of PrimesDiophantine Representation of Primes, , MasterMaster’’s Thesis, University of Pennsylvania, s Thesis, University of Pennsylvania, 2003.2003.

Rowland, E., A Natural PrimeRowland, E., A Natural Prime--Generating Generating Recurrence, Recurrence, Journal of Integer SequencesJournal of Integer Sequences, v.11 , v.11 (2008).(2008).