20030804, 06:55  #1  
"Mike"
Aug 2002
3^{2}·883 Posts 
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:
Thanks! 

20030804, 07:27  #2 
Banned
"Luigi"
Aug 2002
Team Italia
2×3×17×47 Posts 
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 
20030804, 09:29  #3 
2^{2}×5^{2}×59 Posts 
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. 
20030804, 23:32  #4 
Sep 2002
2·331 Posts 
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 ) 
20030805, 00:20  #5 
3·13·241 Posts 
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. :? 
20030805, 16:01  #6 
Nov 2002
Vienna, Austria
41 Posts 
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) 
20030920, 02:13  #7 
Bemusing Prompter
"Danny"
Dec 2002
California
2·3^{2}·131 Posts 
50 primes? Wow!
The best formula I knew before that was p(x) = x^{2} + x + 41. 
20030922, 11:54  #8 
Nov 2002
Vienna, Austria
41 Posts 
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 
20041202, 14:27  #9 
Aug 2004
2·3·7 Posts 
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 
20041213, 16:53  #10  
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
Prime progressions.
Quote:
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 

20041215, 19:46  #11  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
Quote:
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(79m) = (79m)*(79m)  79*(79m) + 1601 = 79^2  79m  79m + m^2  79^2 + 79m + 1601 = m^2  79m + 1601 = f(m) Last fiddled with by cheesehead on 20041215 at 19:59 

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