set of formulas that have no prime numbers

16+60n should be included in the second list..

if n=0 => 5

5 is prime. So the set is wrong.

Bye Jack

The formula is a bit buggy around the edges :D

Guaranteed to do that: 2*n.

I

Given p is prime, and m and n are naturals, m*p + n*p will never be prime!

Big deal!

Hi,

Cool table, did you make that one up? :)

Seems to work great.

Here is another one I came up with today:

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

Here's some output of ratios of how many of the formulas for a given n are prime, ie for n=30 below, 0.36666.. or

11 out of the 30 (11/30=0.3666..) formulas produce primes.

The formula with the lowest ratio's produce the most primes, so for n=210 below (product of primes 2*3*5*7), less than a quarter (0.247619048) of the 210 possible formulas produce primes, ie 52 formulas produce primes for n=210.

This ratio should decrease further for bigger products of primes, ie as above:

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

cheers, Jamie

n ratio

1 1 2 1 3 1 4 0.75 5 1 6 0.666666667 7 1 8 0.625 9 0.777777778 10 0.6 11 1 12 0.5 13 1 14 0.571428571 15 0.666666667 16 0.5625 17 1 18 0.444444444 19 1 20 0.5 21 0.666666667 22 0.545454545 23 1 24 0.416666667 25 0.84 26 0.538461538 27 0.703703704 28 0.5 29 1 30 0.366666667 31 1 32 0.53125 33 0.666666667 34 0.529411765 35 0.742857143 36 0.388888889 37 1 38 0.526315789 39 0.666666667 40 0.45 41 1 42 0.357142857 43 1 44 0.5 45 0.577777778 46 0.52173913 47 1 48 0.375 49 0.87755102 50 0.44 51 0.666666667 52 0.5 53 1 54 0.37037037 55 0.763636364 56 0.464285714 57 0.666666667 58 0.517241379 59 1 60 0.316666667 61 1 62 0.516129032 63 0.603174603 64 0.515625 65 0.769230769 66 0.348484848 67 1 68 0.5 69 0.666666667 70 0.385714286 71 1 72 0.361111111 73 1 74 0.513513514 75 0.56 76 0.5 77 0.805194805 78 0.346153846 79 1 80 0.425 81 0.679012346 82 0.512195122 83 1 84 0.321428571 85 0.776470588 86 0.511627907 87 0.666666667 88 0.477272727 89 1 90 0.3 91 0.813186813 92 0.5 93 0.666666667 94 0.510638298 95 0.778947368 96 0.354166667 97 1 98 0.448979592 99 0.626262626 100 0.42 101 1 102 0.343137255 103 1 104 0.480769231 105 0.485714286 106 0.509433962 107 1 108 0.351851852 109 1 110 0.390909091 111 0.666666667 112 0.446428571 113 1 114 0.342105263 115 0.782608696 116 0.5 117 0.632478632 118 0.508474576 119 0.823529412 120 0.291666667 121 0.917355372 122 0.508196721 123 0.666666667 124 0.5 125 0.808 126 0.30952381 127 1 128 0.5078125 129 0.666666667 130 0.392307692 131 1 132 0.325757576 133 0.827067669 134 0.507462687 135 0.548148148 136 0.485294118 137 1 138 0.34057971 139 1 140 0.364285714 141 0.666666667 142 0.507042254 143 0.853146853 144 0.347222222 145 0.786206897 146 0.506849315 147 0.585034014 148 0.5 149 1 150 0.286666667 151 1 152 0.486842105 153 0.640522876 154 0.409090909 155 0.787096774 156 0.326923077 157 1 158 0.506329114 159 0.666666667 160 0.4125 161 0.832298137 162 0.345679012 163 1 164 0.5 165 0.503030303 166 0.506024096 167 1 168 0.303571429 169 0.928994083 170 0.394117647 171 0.643274854 172 0.5 173 1 174 0.33908046 175 0.697142857 176 0.465909091 177 0.666666667 178 0.505617978 179 1 180 0.283333333 181 1 182 0.412087912 183 0.666666667 184 0.489130435 185 0.789189189 186 0.338709677 187 0.86631016 188 0.5 189 0.582010582 190 0.394736842 191 1 192 0.34375 193 1 194 0.505154639 195 0.507692308 196 0.43877551 197 1 198 0.318181818 199 1 200 0.41 201 0.666666667 202 0.504950495 203 0.837438424 204 0.328431373 205 0.790243902 206 0.504854369 207 0.647342995 208 0.471153846 209 0.870813397 210 0.247619048 211 1 212 0.5 213 0.666666667 214 0.504672897 215 0.790697674 216 0.342592593 217 0.838709677 218 0.504587156 219 0.666666667 220 0.377272727 221 0.877828054 222 0.337837838 223 1

Hi,

Do you have the formula for that polynomial?

So now can you calculate a 500 digit prime number and win some money? :D Or prove a prime gap conjecture.

cheers, Jamie

Here is something that might be applicable:

Many mathematicians believe that the true size of large prime gaps is probably considerably larger ? more on the order of (log X)2, an idea

Gaps of size (log X)2 are what would occur if the prime numbers behaved like a collection of random numbers, which in many respects they appear

said. ?We just don?t understand prime numbers very well.?

cheers, Jamie

Hi,

a+840n for n integers, has more integer values for 'a' that produce primes than any other equation up to a+1000n (still testing higher)

630 number of prime producing equations=148 ratio 148/630 = 0.234920634920635

840 number of prime producing equations= 196 ratio 196/840 = 0.233333333333333

840 seems to have the fewest a values that generate primes since 840 / 7 / 5 / 3 / 2 / 2 / 2 = 1

or in other words 840 can be divided completely by primes.

630 7 / 5 / 3 / 3 / 2 = 1

or same thing these primes can be multiplied to generate these special formulas:

840 = 7*5*3*2*2*2 630 = 7*5*3*3*2

cheers, Jamie

For the number 840 above, according to wikipedia:

it is all of these:

So based on what I find for other numbers with the lowest ratios, like 840 has, I will narrow it down to see which of those three are important for this formula.

cheers, Jamie

A quick targeted test for 2310, chosen to check my idea that products of primes will have the fewest prime producing formulas:

2310=2*3*5*7*11

2310 number of prime producing equations= 485 ratio= 0.20995670995671

wikipedia says this is a "primorial" number:

sequence of primorial numbers:

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410,

Interesting that page says: "Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v, Jan

10 2004"

"Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - Lekraj Beedassy, Mar 31 2005"

"Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/Pi^2 < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie, Jun 11 2005"

" The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015 "

Looks like I finally found what I'm doing, assuming this is the matching sequence.

cheers, Jamie

Here is the list of equation coefficients that produce all the primes for 2310 for n=0 to n=infinity

1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103, 107,109,113,127,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197,199, 211,221,223,227,229,233,239,241,247,251,257,263,269,271,277,281,283,289,293,299, 307,311,313,317,323,331,337,347,349,353,359,361,367,373,377,379,383,389,391,397, 401,403,409,419,421,431,433,437,439,443,449,457,461,463,467,479,481,487,491,493, 499,503,509,521,523,527,529,533,541,547,551,557,559,563,569,571,577,587,589,593, 599,601,607,611,613,617,619,629,631,641,643,647,653,659,661,667,673,677,683,689, 691,697,701,703,709,713,719,727,731,733,739,743,751,757,761,767,769,773,779,787, 793,797,799,809,811,817,821,823,827,829,839,841,851,853,857,859,863,871,877,881, 883,887,893,899,901,907,911,919,923,929,937,941,943,947,949,953,961,967,971,977, 983,989,991,997,1003,1007,1009,1013,1019,1021,1027,1031,1033,1037,1039,1049,1051, 1061,1063,1069,1073,1079,1081,1087,1091,1093,1097,1103,1109,1117,1121,1123,1129, 1139,1147,1151,1153,1157,1159,1163,1171,1181,1187,1189,1193,1201,1207,1213,1217, 1219,1223,1229,1231,1237,1241,1247,1249,1259,1261,1271,1273,1277,1279,1283,1289, 1291,1297,1301,1303,1307,1313,1319,1321,1327,1333,1339,1343,1349,1357,1361,1363, 1367,1369,1373,1381,1387,1391,1399,1403,1409,1411,1417,1423,1427,1429,1433,1439, 1447,1451,1453,1457,1459,1469,1471,1481,1483,1487,1489,1493,1499,1501,1511,1513, 1517,1523,1531,1537,1541,1543,1549,1553,1559,1567,1571,1577,1579,1583,1591,1597, 1601,1607,1609,1613,1619,1621,1627,1633,1637,1643,1649,1651,1657,1663,1667,1669, 1679,1681,1691,1693,1697,1699,1703,1709,1711,1717,1721,1723,1733,1739,1741,1747, 1751,1753,1759,1763,1769,1777,1781,1783,1787,1789,1801,1807,1811,1817,1819,1823, 1829,1831,1843,1847,1849,1853,1861,1867,1871,1873,1877,1879,1889,1891,1901,1907, 1909,1913,1919,1921,1927,1931,1933,1937,1943,1949,1951,1957,1961,1963,1973,1979, 1987,1993,1997,1999,2003,2011,2017,2021,2027,2029,2033,2039,2041,2047,2053,2059, 2063,2069,2071,2077,2081,2083,2087,2089,2099,2111,2113,2117,2119,2129,2131,2137, 2141,2143,2147,2153,2159,2161,2171,2173,2179,2183,2197,2201,2203,2207,2209,2213, 2221,2227,2231,2237,2239,2243,2249,2251,2257,2263,2267,2269,2273,2279,2281,2287, 2291,2293,2297,2309

ie for the last one, the equation 2309 + 2310n for n=0 to n=432 produces 166 primes, pretty decent ratio, that's a 38.4% chance for each value to be prime in that range up to n=432, ie primes up to 1million.

All of those equation coefficients produce similar occurrences of the primes, so all primes up to 1million can be calculated with the set of these formulas with about 38.4% +- a couple percent occurrence.

Also someone said before "what is the point of this if you have to calculate all these coefficients first anyway" well for these special numbers, ie 2310, I think there might be a formula to generate the big list of coefficients above, as there is a quite periodic pattern between the coefficient values. Here is the spacing for the above coefficients:

1,1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,2, 4,6,2,10,2,4,2,12,10,2,4,2,4,6,2,6,4,6,6,6,2,6,4,2,6,4,6,8,4,2,4,6,8,6,10,2,4,6,2, 6,6,4,2,4,6,2,6,4,2,6,10,2,10,2,4,2,4,6,8,4,2,4,12,2,6,4,2,6,4,6,12,2,4,2,4,8,6,4, 6,2,4,6,2,6,10,2,4,6,2,6,4,2,4,2,10,2,10,2,4,6,6,2,6,6,4,6,6,2,6,4,2,6,4,6,8,4,2,6, 4,8,6,4,6,2,4,6,8,6,4,2,10,2,6,4,2,4,2,10,2,10,2,4,2,4,8,6,4,2,4,6,6,2,6,4,8,4,6,8, 4,2,4,2,4,8,6,4,6,6,6,2,6,6,4,2,4,6,2,6,4,2,4,2,10,2,10,2,6,4,6,2,6,4,2,4,6,6,8,4,2, 6,10,8,4,2,4,2,4,8,10,6,2,4,8,6,6,4,2,4,6,2,6,4,6,2,10,2,10,2,4,2,4,6,2,6,4,2,4,6,6, 2,6,6,6,4,6,8,4,2,4,2,4,8,6,4,8,4,6,2,6,6,4,2,4,6,8,4,2,4,2,10,2,10,2,4,2,4,6,2,10,2, 4,6,8,6,4,2,6,4,6,8,4,6,2,4,8,6,4,6,2,4,6,2,6,6,4,6,6,2,6,6,4,2,10,2,10,2,4,2,4,6,2, 6,4,2,10,6,2,6,4,2,6,4,6,8,4,2,4,2,12,6,4,6,2,4,6,2,12,4,2,4,8,6,4,2,4,2,10,2,10,6,2, 4,6,2,6,4,2,4,6,6,2,6,4,2,10,6,8,6,4,2,4,8,6,4,6,2,4,6,2,6,6,6,4,6,2,6,4,2,4,2,10,12, 2,4,2,10,2,6,4,2,4,6,6,2,10,2,6,4,14,4,2,4,2,4,8,6,4,6,2,4,6,2,6,6,4,2,4,6,2,6,4,2,4,12

Maybe comparing these spacings with other values besides 2310 can show the overall formula, then if that is found it should be easier to calculate primes. It's possible there is no pattern that can be found in the spacing too, in fact these special numbers, ie 2310 might just have a more noisy spacing of coefficients than the less useful values which have lower prime density.

If anyone sees a pattern in there though let me know :D

cheers, Jamie

Ok,

I realize now that the list of prime generating coefficients for 2310, is *almost* just the sequence of prime digits, except for 169, and some semi-primes 221 in there etc..

So I am pretty sure that if instead of using 2310, I use a REALLY big number with the same properties, ie assuming the important property is

2310 being primiorial:

then the biggest one listed on that page is: 557940830126698960967415390 so that is the new formula to use, and now since it is such a big number I hypothesize that the sequence of prime generating coefficients for it, are going to be a lot closer to the actual sequence of prime numbers, with fewer semi-primes in there..

So now there is at least a formula to calculate the set of coefficients to generate prime numbers at really big scales.

ie:

7 + 557940830126698960967415390n for n=0 to n=infinity

this formula should have really good prime number density (compared to just doing a brute force prime number search) up to maybe

10^30 or something like that. To calculate all primes using that formula it is the sequence of these formulas (except for possible errors of coefficients that are coprimes being excluded)

2 + 557940830126698960967415390n

3 + 557940830126698960967415390n 5 + 557940830126698960967415390n 7 + 557940830126698960967415390n 11 + 557940830126698960967415390n (all primes) + 557940830126698960967415390n

Calculating all the primes up to 557940830126698960967415390 is still a lot of calculating, but then once they are calculated, then even more primes can be found using that primorial set of formulas.

cheers, Jamie

Especially when Y is even.

Hi,

What about 7 + 2310n?

Primorial numbers ie 2310, have more distinct prime factors than any number smaller than it. Yet Y in this case, ie 7 has the common factor (of itself) with 2310.

The best formula to find the most primes I found so far is:

most likelihood of being prime = a prime below the primorial + (primorial)n

cheers, Jamie

Hi,

So given a very large number that you don't if it is prime or not, this algorithm (with arbitrary precision math) should be able to quickly see if the number is prime without doing any factorization? Is that right, and are there other algorithms that can do this?

cheers, Jamie

Hi,

Actually that is even better, I think I made an error in my code to select these functions as prime producing based on the fact that they produce a single prime.. but if that is true that these formulas produce no more than a single prime, then the new formula excluding the single prime is:

NOT_prime = a prime below the primorial + (primorial)n

Can that formula be used to check if a very large number is prime?

ie. try to subtract integer multiples of primorial numbers from a very large number, and if the result is prime then the very large number isn't prime maybe. ie 9247-2310*4=7

so 9247 isn't prime if that formula is true.

cheers, Jamie

less than this number divides the original number evenly, if not you have a prime. This may be considered brute force. however.

Hi,

Actually that above formula I wrote is wrong, ie for

Y + 2310n, all prime producing functions (that produce more than a single prime) have Y as a prime larger than 11, since 2310 = 2*3*5*7*11.

So my formula before is correct, Y tends to be always prime the bigger then primorial number is, but for Y less than the primorial largest prime factor (ie 11 for 2310) then there is only one prime produced, but larger primes than the largest primorial factor produce many primes.

ie for 13 + 2310n, there are 164 primes produced for n up to 432

164/432=38% primes

cheers, Jamie

Also for 13 + 2310n, there are 1370 primes produced for n up to 4329.

1370/4329=31.6%

cheers, Jamie