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Old 2003-08-04, 06:55   #1
Xyzzy
 
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"Mike"
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Default Prime progressions...

On one of the mailing lists lately there was some talk of prime number progressions that were described as equations... I was wondering what method people use to find these and what the largest might be?

Here is an example from the list...

Quote:
Just in the last week Gary Chaffey reported that the equation 2*x2-88*x+997 generates 51 primes from x=0 to x= 50
I think it might be fun for us to talk about this here... (At least I'm really interested!)

Thanks!
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Old 2003-08-04, 07:27   #2
ET_
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"Luigi"
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Try a search with the keywords "Ulam spiral": there are plenty of sites that try to implement such an equation, but AFAIK no one has still found "the" equation :-)

Luigi
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Old 2003-08-04, 09:29   #3
TTn
 

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Better yet, draw the ulam spiral on graph paper, using only odd numbers.
Our beloved Mersenne primes fall directly along the diagonal. :D

Hmm here is a wacky equation for that diagonal.


__n
\
/__ 8(2n+1)
n=0
minus one.
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Old 2003-08-04, 23:32   #4
dsouza123
 
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Are there any rules, algorithms, heuristics for eliminating numbers or ranges of numbers on the diagonal ?
( Other than doing a TF/LL or under a certain range have all been tested so they can be excluded )
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Old 2003-08-05, 00:20   #5
TTn
 

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I know of none yet drawn, but a real smart mathematician could easily return the probability of n, being prime in the above. You could then mix the current probability of mersenne primes with this, to get a general hybrid prob.
Though all Mn, with n odd appear on the diagonal, so I'm not sure any of this will help... well maybe it could then in that case.
:?
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Old 2003-08-05, 16:01   #6
koal
 
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Hi there!

Somebody proved that a polynomial never will produce just primes. But if I can provide a function, that produces the first 25 primes (those below 100) ...

y = ( -1961755 x^24 + 626211420 x^23 - 94460687338 x^22 + 8955156636096 x^21 - 598606037462125 x^20 + 30003592036428780 x^19 - 1170739327278578728 x^18 + 36446058560245549776 x^17 - 920325981234349785205 x^16 + 19064007207309706990260 x^15 - 326336795367642933428338 x^14 + 4636387497021856699615296 x^13 - 54768570875443768152863635 x^12 + 537643906122042660114998340 x^11 - 4373844384977844634682276188 x^10 +29335829725530079227920667696 x^9 - 160950424593447516378672853840 x^8 + 714365040902901226039992880320 x^7 - 2525953621133868833652063283008 x^6 + 6966549186063153961611970043136 x^5 - 14546083395318615094880191933440 x^4 + 22007289038111101518179578490880 x^3 - 22507759972998750778418906726400 x^2 + 13727914564300379016566243328000 x - 3700818363341536479443681280000 ) /
620448401733239439360000

... what stops me to create a function, that provides the first n primes (or even the first n+1)? Okay, I know, the message is "MACHINE STORAGE EXHAUSTED" :?

for x=26, y becomes -1560168 which is unfortunately even, so the prover mentioned before is right, but the function above really produces all primes from 2 to 97. It behaves like a curve, which goes through the 25 points with coordinates (1/2), (2,3), (3,5), ..., (23,83), (24,89), (25,97)!

Are there limits except storage?
Are there any smaller polynomials for the first 25 (or first n) primes?

Koal 8)
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Old 2003-09-20, 02:13   #7
ixfd64
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50 primes? Wow!

The best formula I knew before that was p(x) = x2 + x + 41.
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Old 2003-09-22, 11:54   #8
koal
 
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The numbers are

997, 911, 829, 751, 677, 607, 541, 479, 421, 367, 317, 271, 229, 191, 157, 127, 101, 79, 61, 47, 37, 31, 29 , 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997 , 1087, 1181, 1279, 1381, 1487, 1597

As you can see, 22 of them are duplicate, which reduces the number of unique primes to 29.

x2 + x + 41 => 40 unique primes
x2 - 79 + 1601 => 80 primes, 40 of them ar duplicates
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Old 2004-12-02, 14:27   #9
synergy
 
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Cool Not strictly polynomials but...

Though they aren't strictly polynomials (I don't know if you say my previous posts), these will give a lot of early primes without composites...

105 - 2^x
15 - 2^x
45 - 2^x
70 - 3^x
75 - 2^x

The "plus" version of these gives alot of them, also.
Aaron
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Old 2004-12-13, 16:53   #10
mfgoode
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Cool Prime progressions.

Quote:
Originally Posted by koal
The numbers are

997, 911, 829, 751, 677, 607, 541, 479, 421, 367, 317, 271, 229, 191, 157, 127, 101, 79, 61, 47, 37, 31, 29 , 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997 , 1087, 1181, 1279, 1381, 1487, 1597

As you can see, 22 of them are duplicate, which reduces the number of unique primes to 29.

x2 + x + 41 => 40 unique primes
x2 - 79 + 1601 => 80 primes, 40 of them ar duplicates


You have made a typographical error in the function in the last line.
Do you mean the function I posted in Thread 'Prime generating polynomials' ? which is f(n) = n^2 -79n +1601

Which function do your duplicate primes refer too ? They certainly dont belong to the function f(n) = n^2 -79n +1601
This function has 80 distinct primes.

Please clarify

Mally






'\\\\\\\\\/l
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Old 2004-12-15, 19:46   #11
cheesehead
 
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Quote:
Originally Posted by mfgoode
You have made a typographical error in the function in the last line.
Do you mean the function I posted in Thread 'Prime generating polynomials' ? which is f(n) = n^2 -79n +1601
I'm pretty sure that's what koal meant.

Quote:
Which function do your duplicate primes refer too ? They certainly dont belong to the function f(n) = n^2 -79n +1601
This function has 80 distinct primes.
Nope. Only 40 distinct prime values for the range n = 0 through 79. (Though who knows how many for n outside that range!) :)

The function is symmetrical around the line n = 39.5

f(0) = 0 - 0 + 1601 = 1601
f(79) = 79^2 - 79^2 + 1601 = 1601 = f(0)

f(39) = 39*39 - 79*39 +1601 = -40*39 +1601 = 41
f(40) = 40*40 -79*40 + 1601 = -39*40 + 1601 = 41 = f(39)

f(79-m) = (79-m)*(79-m) - 79*(79-m) + 1601
= 79^2 - 79m - 79m + m^2 - 79^2 + 79m + 1601
= m^2 - 79m + 1601
= f(m)

Last fiddled with by cheesehead on 2004-12-15 at 19:59
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