set of formulas for computing all prime numbers

It is similar to a primorial number for having at least one of all the prime factors up to 23. Wolfram alpha doesn't compute the divisors for it unfortunately, too big I guess.

Ok but also it has a lot of divisors, wolfram alpha gave this:

"divisors 5,283,186,672,439,900"

{1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 373,

700, 746, 907, 1492, 1814, 1865, 2611, 3628, 3730, 4535, 5222, 6349, 7460, 9070, 9325, 10444, 12698, 13055, 18140, 18650, 22675, 25396, 26110, 31745, 37300, 45350, 52220, 63490, 65275, 90700, 126980, 130550, 158725, 261100, 317450, 338311, 634900, 676622, 1353244, 1691555, 2368177, 3383110, 4736354, 6766220, 8457775, 9472708, 11840885, 16915550, 22309087, 23681770, 33831100, 44618174, 47363540, 59204425, 89236348, 111545435, 118408850, 156163609, 223090870, 236817700, 312327218, 446181740, 557727175, 624654436, 780818045, 1115454350, 1561636090, 2230908700, 3123272180, 3904090225, 7808180450, 8321289451, 15616360900, 16642578902, 20234341909, 33285157804, 40468683818, 41606447255, 58249026157, 80937367636, 83212894510, 101171709545, 116498052314, 141640393363, 166425789020, 202343419090, 208032236275, 232996104628, 283280786726, 291245130785, 404686838180, 416064472550, 505858547725, 566561573452, 582490261570, 708201966815, 832128945100, 1011717095450, 1164980523140, 1416403933630, 1456225653925, 2023434190900, 2832807867260, 2912451307850, 3541009834075, 5824902615700, 7082019668150, 7547409532057, 14164039336300, 15094819064114, 30189638128228, 37737047660285, 52831866724399, 75474095320570, 105663733448798, 150948190641140, 188685238301425, 211327466897596, 264159333621995, 377370476602850, 528318667243990, 754740953205700, 1056637334487980, 1320796668109975, 2641593336219950, 5283186672439900}

Not bad :D

cheers, Jamie

Hi,

So all those divisors can be summed in different ways along with the prime 43,142,746,595,714,191 to sum up to the square of the new prime that is found I think.

example of what I posted before:

" ie for the simple formula 5+6n

6 has unique divisors: 1,2,3,6

and the formula finds the prime number 23 with

5+6(3)

So the number here that is added to the prime is 18, and it has unique divisors:

1, 2, 3, 6, 9, 18

These can be arranged on the X Y graph to give the X Y coordinates of the square for the prime 23, by adding up the different divisors, ie 18+3+2=23 "

cheers, Jamie

hmm the primorial number 6469693230, which is a lot smaller has 1024 divisors and 10 distinct prime factors though according to wolfram alpha. It would be interesting to

the primorial number 200560490130 has 2048 divisors and 11 distinct prime factors.

Hey consecutive primorial numbers have twice as many divisors, and their divisor counts are powers of 2 matching the number of distinct prime factors! :)

ie 2^10 and 2^11 above..

cheers, Jamie

Yep:

"the k-th primorial number (A002110) is the smallest positive integer such that its number of unitary divisors (A034444) is 2^k. Thus the binary orders (A029837) of primorial numbers (A045716) determines the maximal values of unitary divisors in binary order ranges."

Hi,

I have a new conjecture about primes that is similar to Goldbach's conjecture:

's_conjecture

However I am saying that all primes can be written as a sum of a prime number of primes (ie sum of 2,3,5,7.. primes) with the largest always being the prime next to the prime being summed to:

prime(n) = prime(n-1) + next smallest list of primes to sum to prime(n)

count of prime(n-1) + next smallest list of primes to sum to prime(n) = a prime number

ie 23=19+2+2 (prime) and count 3 is prime

Why is this important (maybe)? It can maybe show a difference between sums for primes and for non-primes possibly (not sure)

Also might show a "connection" between consecutive primes, in how new larger primes are created using the smaller primes.

It is possible to strengthen the conjecture to being a sum of

2 or 3 primes, but since one of the primes is set as being the next smallest prime that might be too restrictive.

I think if this can be proven it would describe more what primes are..

2=2 (prime) and count 1 is not prime or composite 3=2+1 (prime) and count 2 sum terms is prime 5=3+2 (prime) and count 2 sum terms is prime 7=5+2 (prime) and count 3 sum terms is prime 11=7+2+2 (prime) and count 3 sum terms is prime 13=11+2 (prime) and count 2 sum terms is prime 17=13+2+2 (prime) and count 3 is prime 19=17+2 (prime) and count 2 is prime 23=19+2+2 (prime) and count 3 is prime 29=23+3+3 (prime) and count 3 is prime 31=29+2 ... all the below primes have count that is prime too 37=31+3+3 41=37+2+2 43=41+2 47=43+2+2 53=47+3+3 59=53+3+3 61=59+2 67=61+3+3 71=67+2+2 73=71+2 79=73+3+3 83=79+2+2 89=83+3+3 97=89+5+3 101=97+2+2 103=101+2 107=103+2+2 109=107+2 113=109+2+2 127=113+11+3 131=127+2+2 137=131+3+3 139=137+2 149=139+5+5 151=149+2 157=151+3+3 163=157+3+3 .... 7243=7237+3+3 7247=7243+2+2 7253=7247+3+3 7283=7253+23+7 7283=7253+17+13 7297=7283+11+3 7307=7297+7+3 .... 247538611=247538597+7+7 253893139=253893137+2 .... 981036977-981036967+7+3 981036989-981036977+7+5 981036997-981036989+5+3 981037007-981036997+7+3 981037069-981037007+59+3 981037081-981037069+7+5 981037091-981037081+7+3 .... 899999902783=899999902759+19+5 899999902829=899999902783+43+3 899999902831=899999902829+2 899999902849=899999902831+13+5 899999902853=899999902849+2+2 899999902859=899999902853+3+3 899999902877=899999902859+13+5 899999902891=899999902877+11+3 899999902897=899999902891+3+3 ....

these trillion size primes from:

This wouldn't work for non-primes I don't think since they are spaced only one digit apart, ie these composite numbers can be spaced one apart, and in that case +1 isnt a prime, but otherwise it may be the same as for the primes above.

composite=nextSmallestPrime+prime(s)

899999902892=899999902891+1 899999902893=899999902891+2 899999902894=899999902891+3 899999902895=899999902891+2+2 899999902896=899999902891+5

cheers, Jamie