Prime progressions...

[Xyzzy]

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...

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!

[ET_]

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

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.

[dsouza123]

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 )

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.
:?

[koal]

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)

[ixfd64]

50 primes? Wow!

The best formula I knew before that was p(x) = x² + x + 41.

[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

[synergy]

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

[mfgoode]

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

[cheesehead]

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.

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)