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generalized formula for distribution of prime numbers

Hi,

Here is the prime number distribution for multiples of 3600

the count of primes for a+3600n for n=0 to n=3599

for the first 140 or so values for a (cropped for space)

I found the pattern! :D

if the formula with a produces more than a single prime, then a is ALWAYS either a prime or a product of primes, before I thought it could be a power of primes, but it is actually a product of primes.

ie for a = 17 below the formula is"

17+3600n for n=0 to n=3599

and at the range of the first 10million primes I tested to, this formula produced 690 primes.

The reason I call this a generalized formula, is because a is always a prime or a product of primes, and the 3600 is arbitrary as long as it is a multiple of 6 I think.

cheers, Jamie

a, count

{[0, 0]} {[1, 690]} {[2, 1]} {[3, 1]} {[4, 0]} {[5, 1]} {[6, 0]} {[7, 698]} {[8, 0]} {[9, 0]} {[10, 0]} {[11, 687]} {[12, 0]} {[13, 694]} {[14, 0]} {[15, 0]} {[16, 0]} {[17, 690]} {[18, 0]} {[19, 691]} {[20, 0]} {[21, 0]} {[22, 0]} {[23, 697]} {[24, 0]} {[25, 0]} {[26, 0]} {[27, 0]} {[28, 0]} {[29, 672]} {[30, 0]} {[31, 689]} {[32, 0]} {[33, 0]} {[34, 0]} {[35, 0]} {[36, 0]} {[37, 712]} {[38, 0]} {[39, 0]} {[40, 0]} {[41, 688]} {[42, 0]} {[43, 684]} {[44, 0]} {[45, 0]} {[46, 0]} {[47, 702]} {[48, 0]} {[49, 688]} {[50, 0]} {[51, 0]} {[52, 0]} {[53, 677]} {[54, 0]} {[55, 0]} {[56, 0]} {[57, 0]} {[58, 0]} {[59, 669]} {[60, 0]} {[61, 678]} {[62, 0]} {[63, 0]} {[64, 0]} {[65, 0]} {[66, 0]} {[67, 686]} {[68, 0]} {[69, 0]} {[70, 0]} {[71, 698]} {[72, 0]} {[73, 671]} {[74, 0]} {[75, 0]} {[76, 0]} {[77, 691]} {[78, 0]} {[79, 693]} {[80, 0]} {[81, 0]} {[82, 0]} {[83, 711]} {[84, 0]} {[85, 0]} {[86, 0]} {[87, 0]} {[88, 0]} {[89, 685]} {[90, 0]} {[91, 700]} {[92, 0]} {[93, 0]} {[94, 0]} {[95, 0]} {[96, 0]} {[97, 696]} {[98, 0]} {[99, 0]} {[100, 0]} {[101, 710]} {[102, 0]} {[103, 682]} {[104, 0]} {[105, 0]} {[106, 0]} {[107, 691]} {[108, 0]} {[109, 680]} {[110, 0]} {[111, 0]} {[112, 0]} {[113, 690]} {[114, 0]} {[115, 0]} {[116, 0]} {[117, 0]} {[118, 0]} {[119, 688]} {[120, 0]} {[121, 683]} {[122, 0]} {[123, 0]} {[124, 0]} {[125, 0]} {[126, 0]} {[127, 730]} {[128, 0]} {[129, 0]} {[130, 0]} {[131, 689]} {[132, 0]} {[133, 692]} {[134, 0]} {[135, 0]} {[136, 0]} {[137, 694]} {[138, 0]} {[139, 688]} {[140, 0]} {[141, 0]}

... continues to 3599

Reply to
Jamie M
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This is about a 24.84% prime production rate for each of these formulas up to 10million digits, which is pretty good.

ie for 17+3600n

out of 2777 values for n, giving 2777*3600=10million numbers,

690 of them return a prime. All the other formulas besides for 17 have a similar prime density rate too up to 10million digits.

To get a high density of primes, it is good to increase the multiple higher than 3600, ie to 216000, since then is a smaller percentage of formulas out of the 216000 that have a as a prime or a product of primes.

cheers, Jamie

Reply to
Jamie M

You might be interested in the Dirichlet prime number theorem:

formatting link
's_theorem_on_arithmetic_progressions

Leon

Reply to
Leon Aigret

I ran a test using 21600 instead of 3600, so the formulas are:

[prime or prime product below 21600] + 21600n (for n=0 to n=21599)

that list of equations produces all the prime numbers, and the distribution still seems to be about 25% for each equation, up to the tested 10million numbers.

Here is a subset of the equations and how many primes they have:

all formulas for y=a+21600n up to 10million numbers:

cheers, Jamie

a, count {[0, 0]} {[1, 120]} {[2, 1]} {[3, 1]} {[4, 0]} {[5, 1]} {[6, 0]} {[7, 108]} {[8, 0]} {[9, 0]} {[10, 0]} {[11, 112]} {[12, 0]} {[13, 114]} {[14, 0]} {[15, 0]} {[16, 0]} {[17, 112]} {[18, 0]} {[19, 117]} {[20, 0]} {[21, 0]} {[22, 0]} {[23, 121]} {[24, 0]} {[25, 0]} {[26, 0]} {[27, 0]} {[28, 0]} {[29, 124]} {[30, 0]} {[31, 116]} {[32, 0]} {[33, 0]} {[34, 0]} {[35, 0]} {[36, 0]} {[37, 121]}

... big gap ...

{[7530, 0]} {[7531, 116]} {[7532, 0]} {[7533, 0]} {[7534, 0]} {[7535, 0]} {[7536, 0]} {[7537, 121]} {[7538, 0]} {[7539, 0]} {[7540, 0]} {[7541, 120]} {[7542, 0]} {[7543, 110]} {[7544, 0]} {[7545, 0]} {[7546, 0]} {[7547, 124]} {[7548, 0]} {[7549, 117]} {[7550, 0]} {[7551, 0]} {[7552, 0]} {[7553, 109]} {[7554, 0]} {[7555, 0]} {[7556, 0]} {[7557, 0]} {[7558, 0]} {[7559, 116]} {[7560, 0]} {[7561, 119]} {[7562, 0]} {[7563, 0]} {[7564, 0]} {[7565, 0]} {[7566, 0]} {[7567, 122]} {[7568, 0]} {[7569, 0]} {[7570, 0]} {[7571, 117]} {[7572, 0]} {[7573, 124]} {[7574, 0]} {[7575, 0]} {[7576, 0]} {[7577, 119]} {[7578, 0]} {[7579, 124]} {[7580, 0]} {[7581, 0]} {[7582, 0]} {[7583, 115]} {[7584, 0]} {[7585, 0]} {[7586, 0]} {[7587, 0]} {[7588, 0]} {[7589, 113]} {[7590, 0]} {[7591, 119]} {[7592, 0]} {[7593, 0]} {[7594, 0]} {[7595, 0]} {[7596, 0]} {[7597, 116]} {[7598, 0]} {[7599, 0]} {[7600, 0]} {[7601, 118]} {[7602, 0]} {[7603, 116]} {[7604, 0]}

... big gap ...

{[20416, 0]} {[20417, 115]} {[20418, 0]} {[20419, 113]} {[20420, 0]} {[20421, 0]} {[20422, 0]} {[20423, 124]} {[20424, 0]} {[20425, 0]} {[20426, 0]} {[20427, 0]} {[20428, 0]} {[20429, 118]} {[20430, 0]} {[20431, 117]} {[20432, 0]} {[20433, 0]} {[20434, 0]} {[20435, 0]} {[20436, 0]} {[20437, 120]} {[20438, 0]} {[20439, 0]} {[20440, 0]} {[20441, 116]} {[20442, 0]} {[20443, 118]} {[20444, 0]} {[20445, 0]} {[20446, 0]} {[20447, 105]} {[20448, 0]} {[20449, 121]} {[20450, 0]} {[20451, 0]} {[20452, 0]} {[20453, 116]} {[20454, 0]} {[20455, 0]} {[20456, 0]} {[20457, 0]} {[20458, 0]} {[20459, 112]} {[20460, 0]} {[20461, 125]} {[20462, 0]} {[20463, 0]} {[20464, 0]} {[20465, 0]} {[20466, 0]} {[20467, 121]} {[20468, 0]} {[20469, 0]} {[20470, 0]} {[20471, 118]} {[20472, 0]} {[20473, 113]} {[20474, 0]} {[20475, 0]} {[20476, 0]} {[20477, 114]} {[20478, 0]} {[20479, 115]} {[20480, 0]} {[20481, 0]} {[20482, 0]} {[20483, 120]} {[20484, 0]} {[20485, 0]} {[20486, 0]} {[20487, 0]} {[20488, 0]} {[20489, 116]} {[20490, 0]} {[20491, 114]} {[20492, 0]} {[20493, 0]}

... big gap ...

{[21580, 0]} {[21581, 118]} {[21582, 0]} {[21583, 116]} {[21584, 0]} {[21585, 0]} {[21586, 0]} {[21587, 124]} {[21588, 0]} {[21589, 123]} {[21590, 0]} {[21591, 0]} {[21592, 0]} {[21593, 122]} {[21594, 0]} {[21595, 0]} {[21596, 0]} {[21597, 0]} {[21598, 0]} {[21599, 115]}

end of sequence

Reply to
Jamie M

Hi,

Thanks I'll try to figure it out if its related or a more generalized solution I think it sounds like.

cheers, Jamie

Reply to
Jamie M

Here is an updated formula:

product of list of primes, ie 2*3*5*7 = 210

coefficient+210n

That formula has the fewest integer coefficients of any other values less than 210 to generate all the primes.

Another example with even more dense primes in the formulas:

product of primes ie: 2*3*5*7*11*13*17*19*23*29 =6469693230

integer coefficient+6469693230n

That formula has the fewest integer coefficients of any other values less than 6469693230 to generate all the primes.

See the pattern! :D

Basically if you take the full set of all infinite prime numbers, and take the product of all of them combined, then make a formula with that like the above, you will still have the best ability to generate all the primes with the highest occurrence density.

cheers, Jamie

Reply to
Jamie M

Hi,

Sure thats great but did they show that for very large primorial numbers the coefficients to create all primes are also prime?

ie integer coefficient+6469693230n

integer coefficients are basically all primes below 6469693230 to generate all prime numbers. The bigger the primorial number is the closer match the integer coefficients are to the prime numbers.

cheers, Jamie

Reply to
Jamie M

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