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set of formulas that have no prime numbers

Hi,

1 + 2310n has 1358 primes for n up to 4329

1358/4329 = 31.3% primes

Here's a list in general for the formulas that return primes for Y + 2310n for Y up to 104.

The occurence of primes is pretty even, and as the primorial number size increases, ie instead of

2310 if 9699690 or larger is used etc, then the Y values match the primes more closely, but even for 2310 it is pretty close.

Y, number of primes produced for n up to 4329

{[0, 0]} {[1, 1358]} {[2, 1]} {[3, 1]} {[4, 0]} {[5, 1]} {[6, 0]} {[7, 1]} {[8, 0]} {[9, 0]} {[10, 0]} {[11, 1]} {[12, 0]} {[13, 1370]} {[14, 0]} {[15, 0]} {[16, 0]} {[17, 1388]} {[18, 0]} {[19, 1377]} {[20, 0]} {[21, 0]} {[22, 0]} {[23, 1382]} {[24, 0]} {[25, 0]} {[26, 0]} {[27, 0]} {[28, 0]} {[29, 1372]} {[30, 0]} {[31, 1371]} {[32, 0]} {[33, 0]} {[34, 0]} {[35, 0]} {[36, 0]} {[37, 1365]} {[38, 0]} {[39, 0]} {[40, 0]} {[41, 1395]} {[42, 0]} {[43, 1417]} {[44, 0]} {[45, 0]} {[46, 0]} {[47, 1380]} {[48, 0]} {[49, 0]} {[50, 0]} {[51, 0]} {[52, 0]} {[53, 1383]} {[54, 0]} {[55, 0]} {[56, 0]} {[57, 0]} {[58, 0]} {[59, 1403]} {[60, 0]} {[61, 1370]} {[62, 0]} {[63, 0]} {[64, 0]} {[65, 0]} {[66, 0]} {[67, 1384]} {[68, 0]} {[69, 0]} {[70, 0]} {[71, 1354]} {[72, 0]} {[73, 1422]} {[74, 0]} {[75, 0]} {[76, 0]} {[77, 0]} {[78, 0]} {[79, 1388]} {[80, 0]} {[81, 0]} {[82, 0]} {[83, 1387]} {[84, 0]} {[85, 0]} {[86, 0]} {[87, 0]} {[88, 0]} {[89, 1404]} {[90, 0]} {[91, 0]} {[92, 0]} {[93, 0]} {[94, 0]} {[95, 0]} {[96, 0]} {[97, 1405]} {[98, 0]} {[99, 0]} {[100, 0]} {[101, 1391]} {[102, 0]} {[103, 1413]} {[104, 0]}

Here is the full list of Y that produce primes in Y = 2310n

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

cheers, Jamie

Reply to
Jamie M
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[snip]

According to wikipedia:

formatting link

"the average gap between consecutive prime numbers among the first N integers is roughly log(N)"

So even though the distribution of primes goes down proportional to the natural log of N, if the above formula with a primorial number is used, then the coefficients Y distribution goes down at roughly the log(N) too, since they approach being the prime numbers (I think) with large primorial numbers.

So this means for big numbers, even though the primes are spaced farther apart (from the prime number theorem), if the primorial equations are used, there are also proportionately fewer Y that contain all the prime numbers as a percentage of all the possible Y coefficients up to the primorial number - 1.

ie for primorial 9699690

Y + 9699690n

Y's max value for producing primes is at most 1 less than 9699690, and the values of Y are almost all primes themselves, so if a larger primorial number is used, ie 200560490130, the number of Y producing equations will grow the same as the prime number theorem essentially, however all Y will grow much larger.

So there is always a lower percentage of equations that have to be searched for primes as the primorial number increases in size, that is proportionate to the increased spacing of the prime numbers.

The growth rate of prime producing Y's is log(N) of the growth rate of the primordial number size I think.

cheers, Jamie

Reply to
Jamie M

Hi,

Basically for a primorial number about this size:

100,000,000,000,000,000

there will be approximately 2,623,557,157,654,233 values for Y that return all the prime numbers.

So about 2.6% of the possible values of Y return all the primes, and the other 97.4% of the values for Y have no primes I think.

cheers, Jamie

Reply to
Jamie M

This gets better for larger primorial numbers too, ie for a primorial with size somewhere around:

1,000,000,000,000,000,000,000,000

there are only 18,435,599,767,349,200,867,866 primes with a value lower than that number, and so a comparably sized primorial:

117,288,381,359,406,970,983,270

about 10x smaller, will have AT MOST

18,435,599,767,349,200,867,866 values of Y that produce primes (actually it will be quite a bit less as only primes below the primorial size are required).

But at the most it is about 1.5% of the possible values of Y that actually produce all the primes, and the other 98.5%+ values of Y below the primorial don't have any primes, if my idea is correct.

So the percentage of prime producing Y's went from about 2.6% down to 1.5% (actually even less) as the primorial size increased, and this matches the prime number theorem too, so these small subset of formulas should maintain their prime density distribution for big primes.

cheers, Jamie

Reply to
Jamie M

For 13 + 2310n, there are 11997 primes produced for n=0 up to n=43290

11997/43290=27.7%

That is a 27.7% chance of finding primes with 13 + 2310n up to n=43290

cheers, Jamie

Reply to
Jamie M

Ok here is a good result:

For the primorial number 510510

Y + 510510n

for all Y's where Y + 510510n for n=0 to n=infinity will have more than one prime number, almost all the Y's themselves are prime too as I had said before, and the first Y that isn't prime is 19^2. This number

19 is the next prime above the primorial number 510510 used to create the next primorial number 9699690

ie.

2*3*5*7*11*13*17 = 510510 2*3*5*7*11*13*17*19 = 510510*19 = 9699690

So it looks like the exact pattern of Y's that produce primes is totally predictable and also is still proportional to just the count of prime numbers below the primorial number used.

Here are the list of Y's for Y + 510510n that produce primes. The first non-prime in the sequence is 19^2 (361) as stated above related to the primorial numbers.

I haven't checked past 361 to see what other non-primes there are, but it seems pretty obvious that the values of Y tend towards being all primes for large primorial numbers.

values of Y in Y + 510510n that produce primes 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 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 361 367 373 379 383 389 397 401 409 419 421 431 433 437 439 443 449 457 461 463 467 479 487 491 499 503 509 521

cheers, Jamie

Reply to
Jamie M

Any Y that have prime factors that are larger than the primorial factors will also be included in the list for Y in the formula

Y + primorial(n)

So the primorial number has to be HUGE in order to have Y be a list of only primes.

Here is the list of all values of Y I found in Y + 510510n up to ~1000 that aren't prime (but all appear to be semiprimes, with two prime factors and both of which are greater than 19, so the next primorial number substituted in the formula would not have them in the list of values of Y if they have a 19 prime factor.

361,437,529,589,667,703,713,779,817,841,851,893,899,943,961,989,1007,1073

So for the generalized formula to be correct that there are only primes as values of Y to find all primes for Y + primorial(n), the primorial number has to be infinite and in fact a product of all the primes in the list Y. If the list of Y is the infinite list of primes, then the primorial number is the product of all the primes in that list.

the full formula that will give all primes is:

infinite list of primes + product of infinite list of primes(n) for n=0 to n=infinity

"product of infinite list of primes" is the biggest primorial number.

cheers, Jamie

Reply to
Jamie M

Looks like a bad formula, even I can see that :D But was fun.

Reply to
Jamie M

Not a totally useless formula I guess, for non infinite sized primorial numbers, just all semi primes with factors above the primorial number have to be added to the list.

Reply to
Jamie M

Cool work.

cheers, Jamie

Reply to
Jamie M

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