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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).