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what is this pattern in coprimes of primorials 30 and 210?

Hi,

For the coprimes of 30:

1,7,11,13,17,19,23,29 I organized them into two columns like this, where the sum of each row is 30:

1 29

7 23 11 19 13 17

Then took the product of each row:

29 161 209 221

Then I took the difference of these:

161-29=132 209-161=48 221-209=12

And noticed that 132,48 and 12 all have greatest common divisor 12.

I also did the same for the coprimes of 210:

1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113, 121,127,131,137,139,143,149,151,157,163,167,169,173,179,181,187,191,193,197,199,209,

I organized them into two columns like this, where the sum of each row is 210:

1 209 11 199 13 197 17 193 19 191 23 187 29 181 31 179 37 173 41 169 43 167 47 163 53 157 59 151 61 149 67 143 71 139 73 137 79 131 83 127 89 121 97 113 101 109 103 107

Then took the product of each row:

209 2189 2561 3281 3629 4301 5249 5549 6401 6929 7181 7661 8321 8909 9089 9581 9869 10001 10349 10541 10769 10961 11009 11021

Then I took the difference of these:

ie: 2189-209=1980

1980 372 720 348 672 948 300 852 528 252 480 660 588 180 492 288 132 348 192 228 192 48 12

And noticed that all these also have greatest common divisor of 12, just like for primorial 30.

So I am guessing that all primorials have the same pattern but not sure.

The question I have is where does this greatest common divisor of 12 come from?

Here is a picture of the spreadsheet I used:

formatting link

cheers, Jamie

Reply to
Jamie M
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Take all this number theory to a serious math group, where it will get the reception that it deserves.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics
Reply to
John Larkin

There are lots of smart math people here and on sci.math, probably most not as interested in prime numbers as I currently am, but to each their own hehe

Reply to
Jamie M

Keep your silly ideas to yourself. Your insights into prime numbers really aren't worth publishing, and if you told smart math people about them they'd make their excuses and leave.

It's not as actively dangerous as recommending the drinking of unpasteurised milk, but it's at least as fatuous.

--
Bill Sloman, Sydney
Reply to
bill.sloman

and if you told smart math people about them they'd make their excuses and leave.

least as fatuous.

That's great info Bill, if you can focus a bit more beyond insults and publishing papers, I'd appreciate if you can take a look to try to figure out the question I posted (not at all related to publishing a paper, as I am just learning about prime numbers currently)

I think perhaps you have not enough mental capacity to figure it out just like your lack of logic in your other arguments, but give it a shot, math is fun.

cheers, Jamie

Reply to
Jamie M

For the people in sci.math yes. But not for the average person in sci.electronics.design. When they are interested, they can always subscribe sci.math. So please stop posting in sci.electronics.design.

Reply to
Rob

and if you told smart math people about them they'd make their excuses and leave.

as fatuous.

Here is the answer Bill by the way (from sci.math):

Let's call the primorial p and the "coprimes" m and n. Then the

products become m (p - m) and n (p - n) and the difference of the two

can be rewritten as d = (p - m - n) (m - n).

Since all "coprimes" are odd, their difference, their sum, and

p - m - n are all even, so d is a multiple of 4.

Each "coprime" is of the type 3 k + 1 or 3 k + 2. If m and n are of

the same type then m - n is divisible by 3. If they are of different

types then m + n and p - m - n are divisible by 3.

So in both cases d is a multiple of 3, which in combiation with the

divisibility by 4 means that d is a multiple of 12.

As you can see, the proof takes just some simple algebra and facts

that have been mentioned here before.

Reply to
Jamie M

Nah I post wherever I want.

Reply to
Jamie M

If you define

f(n) ? Prog v ? SELECT(GCD(k, n) = 1, k, 1, n) l ? DIM(v)

u ? VECTOR(w?(i + 1) - w?i, i, 1, l/2 - 1) RETURN GCD(u)

then f(#p) = 12 for almost p = 5, 7, 11, 13 and 17, where #p is the primorial of p, i.e.

#p = Product(k, k prime 1 f(6*m) = 12, m > 1, if m odd (it include primorials) = 24, if m even ....

Particularly, if n = 6*m with m odd, the factors are

(3m-k)(3m+k) = 9m^2 - k^2

with k even, k =/= 0 (mod 3). It is, k = 2 or 4 (mod 6)

Then, two consecutive factors are

9m^2 - (6s + 2)^2, 9m^2 - (6s + 4)^2

and its difference is

(6s + 4)^2 - (6s + 2)^2 =

24s + 12 = 12(2s + 1)

ever multiple of 12.

As the top two products are

9m^2 - 4, 9m^2 - 16

its difference is 12, and then all the differences are multiples of 12.

Perhaps it must be completed in some aspect, but that is the idea

Saludos,

snipped-for-privacy@mundo-r.com

formatting link

Reply to
Ignacio Larrosa Cañestro

If you want to publish your elementary math homework, find someplace where it fits. Prime number theory is interesting in its own right, but not here, and you really aren't addressing any of the interesting bits of it.

What you think reflects your own capacity to think - which isn't impressive. Don't waste our time telling us what you think - you've wasted a lot of bandwidth here demonstrating that you can't think to any useful purpose on a wide variety of subjects.

--
Bill Sloman, Sydney
Reply to
bill.sloman

His toilet training isn't up to much either.

--
Bill Sloman, Sydney
Reply to
bill.sloman

t

lly aren't worth publishing,

rised milk, but it's at least

And this is in some way useful?

The last time I got involved in prime numbers was when Donald Davies wanted to put public key encryption into the Telex system (which was a communicat ing word processor scheme designed to replace Telex/TWX, which actually ran in Sweden and Germany in the 1980's before it got buried by e-mail).

formatting link

Co-primes didn't feature.

--
Bill Sloman, Sydney
Reply to
bill.sloman

you told smart math people about them

in its own right, but not here, and you really aren't addressing any of the interesting bits of it.

think to any useful purpose on a wide variety of subjects.

I saw your message, and for a moment I thought you had given a proof of the question I asked, which three other people chose to prove in three different ways. But of course Bill as typical you only wrote more nonsense and insults.

cheers, Jamie

Reply to
Jamie M

On Thu, 31 Mar 2016 06:07:44 -0700, Jamie M Gave us:

WRONG GROUP, STUPID PUTZ!

You are a Prime Retard. You rank right up there with Donald Trump.

Reply to
DecadentLinuxUserNumeroUno

into the Telex system (which was a communicating word processor scheme designed to replace Telex/TWX, which actually ran in Sweden and Germany in the 1980's before it got buried by e-mail).

They aren't actually called coprimes Bill, they are called totients or relatively prime, or reduced residue numbers.

That is why he put them in quotes because unlike you he can see they aren't coprimes. Don't worry I just figured it out too.

No one said it is useful but does that matter so much? Good to see you asking a question for once though.

cheers, Jamie

Reply to
Jamie M

On Thu, 31 Mar 2016 06:11:59 -0700, Jamie M Gave us:

You have no place, nor the qualifications to assess others' actions here in Usenet, nor anywhere else. Or is it the other Jamie I am thinking of or is that another one of *your* sock puppets?

Still... WRONG GROUP, ASSHOLE!

Reply to
DecadentLinuxUserNumeroUno

If you knew anything about electronics, you would be more careful about blowing a fuse.

Reply to
Jamie M

:

get

eally aren't worth publishing,

eurised milk, but it's at least

12

nted to put public key encryption

Whatever.

After being told.

I quite frequently ask questions, but your reading comprehension is letting you down, as usual.

Reader attention is a resource. If you use it up to no purpose, people will stop reading your stuff. You are a slightly special case, in that your ign orance and fatuous self-confidence is frequently unintentionally comic, and I sometimes scan your stuff for the unintended jokes, and other peoples co mic put-downs of your pretensions, but only if I'm pretty bored.

--
Bill Sloman, Sydney
Reply to
bill.sloman

You are a slightly special case, in that your ignorance and fatuous self-confidence is frequently unintentionally comic, and I sometimes scan your stuff for the unintended jokes, and other peoples comic put-downs of your pretensions, but only if I'm pretty bored.

That's great Bill, whatever floats your boat right!

cheers, Jamie

Reply to
Jamie M

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