prime number distribution multiples of 6

Hi,

I generated a list of all the primes in the first 1billion digits, then counted the spacing between all the primes, and got this result:

"name" is the spacing between consecutive primes, and "result" is the count of how many times that spacing occurred, as you can see from the below, the spacing of 6 is the most common gap between primes, and also there is an oscillation of the spacing, where multiples of 6 spacings always have more primes.

ie for name=6 that is the count of all the primes in the first 1billion digits of numbers with a prime spacing of 6.

How do you explain that there are these peaks in the prime distribution on multiples of 6

There is a nice curve that the distribution of primes generates as well from this data.

{ Name = 1, Count = 1 } { Name = 2, Count = 3424506 } { Name = 4, Count = 3424679 } { Name = 6, Count = 6089791 } { Name = 8, Count = 2695109 } { Name = 10, Count = 3484767 } { Name = 12, Count = 4468957 } { Name = 14, Count = 2464565 } { Name = 16, Count = 1846097 } { Name = 18, Count = 3351032 } { Name = 20, Count = 1824043 } { Name = 22, Count = 1569679 } { Name = 24, Count = 2367474 } { Name = 26, Count = 1119585 } { Name = 28, Count = 1219243 } { Name = 30, Count = 2177991 } { Name = 32, Count = 683896 } { Name = 34, Count = 719531 } { Name = 36, Count = 1171524 } { Name = 38, Count = 548746 } { Name = 40, Count = 648780 } { Name = 42, Count = 954456 } { Name = 44, Count = 389634 } { Name = 46, Count = 334720 } { Name = 48, Count = 577247 } { Name = 50, Count = 328066 } { Name = 52, Count = 245804 } { Name = 54, Count = 410754 } { Name = 56, Count = 211462 } { Name = 58, Count = 181948 } { Name = 60, Count = 371839 } { Name = 62, Count = 115558 } { Name = 64, Count = 118951 } { Name = 66, Count = 216787 } { Name = 68, Count = 88396 } { Name = 70, Count = 125564 } { Name = 72, Count = 126663 } { Name = 74, Count = 62526 } { Name = 76, Count = 55113 } { Name = 78, Count = 105313 } { Name = 80, Count = 53522 } { Name = 82, Count = 37982 } { Name = 84, Count = 78077 } { Name = 86, Count = 27793 } { Name = 88, Count = 28878 } { Name = 90, Count = 58057 } { Name = 92, Count = 19282 } { Name = 94, Count = 17669 } { Name = 96, Count = 31078 } { Name = 98, Count = 16175 } { Name = 100, Count = 16900 } { Name = 102, Count = 22393 } { Name = 104, Count = 10310 } { Name = 106, Count = 8719 } { Name = 108, Count = 15459 } { Name = 110, Count = 9065 } { Name = 112, Count = 7139 } { Name = 114, Count = 10892 } { Name = 116, Count = 4710 } { Name = 118, Count = 4502 } { Name = 120, Count = 9621 } { Name = 122, Count = 2975 } { Name = 124, Count = 3136 } { Name = 126, Count = 5863 } { Name = 128, Count = 2043 } { Name = 130, Count = 2813 } { Name = 132, Count = 3510 } { Name = 134, Count = 1487 } { Name = 136, Count = 1297 } { Name = 138, Count = 2589 } { Name = 140, Count = 1516 } { Name = 142, Count = 953 } { Name = 144, Count = 1555 } { Name = 146, Count = 668 } { Name = 148, Count = 724 } { Name = 150, Count = 1486 } { Name = 152, Count = 506 } { Name = 154, Count = 583 } { Name = 156, Count = 798 } { Name = 158, Count = 306 } { Name = 160, Count = 360 } { Name = 162, Count = 534 } { Name = 164, Count = 247 } { Name = 166, Count = 182 } { Name = 168, Count = 443 } { Name = 170, Count = 198 } { Name = 172, Count = 158 } { Name = 174, Count = 255 } { Name = 176, Count = 128 } { Name = 178, Count = 106 } { Name = 180, Count = 200 } { Name = 182, Count = 80 } { Name = 184, Count = 89 } { Name = 186, Count = 101 } { Name = 188, Count = 33 } { Name = 190, Count = 63 } { Name = 192, Count = 74 } { Name = 194, Count = 32 } { Name = 196, Count = 41 } { Name = 198, Count = 73 } { Name = 200, Count = 28 } { Name = 202, Count = 23 } { Name = 204, Count = 46 } { Name = 206, Count = 13 } { Name = 208, Count = 18 } { Name = 210, Count = 48 } { Name = 212, Count = 12 } { Name = 214, Count = 11 } { Name = 216, Count = 15 } { Name = 218, Count = 6 } { Name = 220, Count = 12 } { Name = 222, Count = 11 } { Name = 224, Count = 6 } { Name = 226, Count = 6 } { Name = 228, Count = 3 } { Name = 230, Count = 3 } { Name = 232, Count = 1 } { Name = 234, Count = 13 } { Name = 236, Count = 5 } { Name = 238, Count = 1 } { Name = 240, Count = 5 } { Name = 242, Count = 4 } { Name = 244, Count = 2 } { Name = 246, Count = 4 } { Name = 248, Count = 4 } { Name = 250, Count = 4 } { Name = 252, Count = 1 } { Name = 260, Count = 1 } { Name = 276, Count = 1 } { Name = 282, Count = 1 }

cheers, Jamie

Reply to
Jamie M
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Hi,

In case you don't understand what I'm talking about:

{ Name = 1, Count = 1 } { Name = 2, Count = 3424506 } { Name = 4, Count = 3424679 } { Name = 6, Count = 6089791 } { Name = 8, Count = 2695109 } { Name = 10, Count = 3484767 } { Name = 12, Count = 4468957 } { Name = 14, Count = 2464565 } { Name = 16, Count = 1846097 } { Name = 18, Count = 3351032 } { Name = 20, Count = 1824043 }

ie for name=6 count is greater than name =4 and name =8 (the ones before and after) and all name=6multiples (ie 12,18,24,30) have a count greater than the one right before and after too count is how many times primes occurred with a gap of "name" in the sequence.

cheers, Jamie

Reply to
Jamie M

The prime number 6 thing is old news.

formatting link

Reply to
Wanderer

Hi,

Thanks, here is the prime number distribution for multiples of 60

the count of primes for a+60n for n=0 to n=59

a, count

{[0, 0]} {[1, 4863]} {[2, 1]} {[3, 1]} {[4, 0]} {[5, 1]} {[6, 0]} {[7, 4899]} {[8, 0]} {[9, 0]} {[10, 0]} {[11, 4916]} {[12, 0]} {[13, 4919]} {[14, 0]} {[15, 0]} {[16, 0]} {[17, 4917]} {[18, 0]} {[19, 4919]} {[20, 0]} {[21, 0]} {[22, 0]} {[23, 4928]} {[24, 0]} {[25, 0]} {[26, 0]} {[27, 0]} {[28, 0]} {[29, 4888]} {[30, 0]} {[31, 4943]} {[32, 0]} {[33, 0]} {[34, 0]} {[35, 0]} {[36, 0]} {[37, 4913]} {[38, 0]} {[39, 0]} {[40, 0]} {[41, 4894]} {[42, 0]} {[43, 4905]} {[44, 0]} {[45, 0]} {[46, 0]} {[47, 4892]} {[48, 0]} {[49, 4868]} {[50, 0]} {[51, 0]} {[52, 0]} {[53, 4911]} {[54, 0]} {[55, 0]} {[56, 0]} {[57, 0]} {[58, 0]} {[59, 4917]}

Reply to
Jamie M

[snip]

It provides one means to do a quick lookup table of bits to compute "isprime" quickly for a given number P since dividing P-1 by 3 maps all the numbers apart from 2 that might be prime onto consecutive integers.

P = 3 -> (P-1)/3 = 0 P = 6n-1 -> (P-1)/3 = 2n-1 P = 6n+1 -> (P-1)/3 = 2n

I once used this trick on a puzzle in SciAm Mathematical games using an Epson V30 PC. I came third beaten by a Cray and CDC using brute force.

Having a series of 65536 bit (aka 8kb) long lookup tables for primality speeded up the problem by orders of magnitude. Back in the days when memory was expensive and 640k was thought to be "enough for anybody".

--
Regards, 
Martin Brown
Reply to
Martin Brown

It's one of my childhood rediscoveries. When you look at a calender, all the primes fall in the diagonals below 5 and 7. Neat but known.

Reply to
Wanderer

It's the mark of a crank to start repeated threads on the same topic in a short space of time.

Sylvia.

Reply to
Sylvia Else

No need to cross post that is there Sylvia?

Reply to
Jamie M

I just hit "follow up".

Sylvia.

Reply to
Sylvia Else

You hit that mark right on the spot. Sad that you quoted and reposted his inane horseshit, however.

Reply to
DecadentLinuxUserNumeroUno

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