Distributions of Primes

[steve_bank]

Eureka!!! WE have been eagerly awaiting a new square wave function.

I have no idea what you are doing. I can tell you the utility of the Heaviside step function, the unit step, has wide mathematical application in electric systems. The unit step is mathematically convoluted with a system transfer function to get the step response which is important for several reasons. An electrical step signal is widely used to test physical systems.

The other two major matmatical stimuli are the sine wave and the unit impulse.

A different approach for you would be to look at it as a digital simulation instead of closed direct mathematics.

A sqare wave becomes an array of ones and zeros. Point by point the digital square wave is input to a digital model of the distribution of primes.

Square waves or more generally rectangular waves usualy in reduce to a form of Fourier series.

Another thing to consider is the dc average of the square wave which may cause problems in your math. The average value of a square wave from 0 to 1 is 0.5. From -1 to 1 the average is 0.

You can try a search on La Place transform pair square wave. You might find a differential equation.

 

[steve_bank]

Exponentials can be used to create the rise and fall edges of a square wave if you wnat a continuous function.

Vary tau to adjust the rate of chnage of the edges. Vary k for amplitude.


n = 1000
tmax = 1.
dt = tmax/n
t = 0:dt:tmax
k = 1.
tau = .01
for i = 1:length(t)
yon(i) = 1. - k*%e^(-t(i)/tau)
yoff(i) = k*%e^(-t(i)/tau)
end

plot2d(t,yon)

 

[AdamWho]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

 

[Jarhyn]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

 

[AdamWho]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

p2 = 24k + 1
p2 - 1 = 24k
(p+1)(p-1) = 24k

note: 24 = 2*3*4
since (p-1), p, (p+1) are consecutive numbers

p cannot be a multiple of 2,3 or 4

But (p-1) and (p+1) have to a factor of 2

And (p-1) and/or (p+1) have to have factors of 3 and 4

Therefor (p-1)(p+1) is a multiple of 24

(p-1)(p+1) = 24k

p^2 = 24k +1

 

[Jarhyn]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

p2 = 24k + 1
p2 - 1 = 24k
(p+1)(p-1) = 24k

note: 24 = 2*3*4
since (p-1), p, (p+1) are consecutive numbers

p cannot be a multiple of 2,3 or 4

But (p-1) and (p+1) have to a factor of 2

And (p-1) and/or (p+1) have to have factors of 3 and 4

Therefor (p-1)(p+1) is a multiple of 24

(p-1)(p+1) = 24k

p^2 = 24k +1

I asked for a link to a proof.

I don't have the time or inclination to sort out your post for logic errors given the fact it is riddle with typos.

 

[Jarhyn]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

p2 = 24k + 1
p2 - 1 = 24k
(p+1)(p-1) = 24k

note: 24 = 2*3*4
since (p-1), p, (p+1) are consecutive numbers

p cannot be a multiple of 2,3 or 4

But (p-1) and (p+1) have to a factor of 2

And (p-1) and/or (p+1) have to have factors of 3 and 4

Therefor (p-1)(p+1) is a multiple of 24

(p-1)(p+1) = 24k

p^2 = 24k +1

I asked for a link to a proof.

I don't have the time or inclination to sort out your post for logic errors given the fact it is riddle with typos.

...

It's not an unreasonable request for an established proof on a site or resource devoted to such, given this is a claim I have not seen. I would like to see more indepth discussion on it.

 

[AdamWho]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

p2 = 24k + 1
p2 - 1 = 24k
(p+1)(p-1) = 24k

note: 24 = 2*3*4
since (p-1), p, (p+1) are consecutive numbers

p cannot be a multiple of 2,3 or 4

But (p-1) and (p+1) have to a factor of 2

And (p-1) and/or (p+1) have to have factors of 3 and 4

Therefor (p-1)(p+1) is a multiple of 24

(p-1)(p+1) = 24k

p^2 = 24k +1

I asked for a link to a proof.

I don't have the time or inclination to sort out your post for logic errors given the fact it is riddle with typos.

...

It's not an unreasonable request for an established proof on a site or resource devoted to such, given this is a claim I have not seen. I would like to see more indepth discussion on it.

This is elementary. Learn to talk to your peers. I have no interest in ever interacting with you again.

 

[Jokodo]

Off topic...

The square of every prime (p>4) is a multiple of 24, plus 1

p2 = 24k +1

Actually not off topic.

24 is related to a primorial by a factor of 4.

Is there a point as well where this becomes true of other multiples of primorials, on some other range? I'm curious of the relationship. And whether you can link me to a proof.

And now I want to take a look at the relationships between lower primorial multiples and their neighbors at p<=4.

p2 = 24k + 1
p2 - 1 = 24k
(p+1)(p-1) = 24k

note: 24 = 2*3*4
since (p-1), p, (p+1) are consecutive numbers

p cannot be a multiple of 2,3 or 4

But (p-1) and (p+1) have to a factor of 2

And (p-1) and/or (p+1) have to have factors of 3 and 4

Therefor (p-1)(p+1) is a multiple of 24

(p-1)(p+1) = 24k

p^2 = 24k +1

I asked for a link to a proof.

I don't have the time or inclination to sort out your post for logic errors given the fact it is riddle with typos.

That was a proof.

Among any three consecutive integers, such as p-1, p, p+1, one must be a multiple of 3; for p>3, the choice is reduced to p-1 and p+1 since p, by definition is not a multiple of 3.

Among any two consecutive even integers, such as p-1 and p+1 for any p>2, one must be a multiple of 4, while both are, by definition, multiples of 2.

So (p-1) * (p+1), for p>3, will always be the multiplication of a multiple of 2 with a multiple of 4, with one of them also being a multiple of 3. The product is thus both a multiple of 8 and a multiple of 3. That means it has to be a multiple of 24.

Since (p-1)*(p+1) == p² - 1, or equivalently p² = (p-1)*(p+1) + 1, it follows that p² for p>3 is a multiple of 24, plus 1.

 

[Jokodo]

Alternatively, you just base your proof on the fact that any prime>3 is representable as 6k+/-1, in addition to the simple fact that odd +/-/* odd = even, as does even +/-/* even

for odd k and p=6k-1: (6k - 1)² = 36k² - 12k + 1 = 36 times an odd number minus 12 times an odd number = 12 times an odd number minus 12 times (another) odd number = 12 times some even number = 24 times an integer, plus one

for even k and p=6k-1: (6k - 1)² = 36k² - 12k + 1 = 36 times an even number minus 12 times an (other) even number = 12 times an eveb number minus 12 times (another) even number = 12 times some even number = 24 times an integer, plus one


for odd k and p=6k+1: 36k² +12k + 1 = 36 times an odd number plus 12 times an odd number = 12 times an odd number plus 12 times (another) odd number = 12 times some even number = 24 times an integer, plus one

for even k and p=6k+1: (6k - 1)² = 36k² + 12k + 1 = 36 times an even number plus 12 times an (other) even number = 12 times an even number plus 12 times (another) even number = 12 times some even number = 24 times an integer, plus one

That exhausts the possibilities

 

[Jokodo]

Of course, odd*odd=odd. The exposition makes clear that's what I was using.

 

[Jarhyn]

...

Of course, odd*odd=odd. The exposition makes clear that's what I was using.

Well, it's not a link but it does provide the actual discussion I was looking for in clear terms, so thank you.

At some point I looked it up myself (the origin of the row being that I couldn't because I was on mobile and very busy and would be for much of the day), and asked the question "well, what's the deal with K, how does it grow?"

I interested to see that when I looked up the progression of K that they were in fact a progression of pentagonal numbers.

My real question is what is it about k=22,... That makes these pentagonal numbers lie about whether their result in k^2/24+1/24=**** that makes this number nonprime?

I'm going to look into the relationship of these "liars" and why they happen later.

I also found a paper from 2019 about a proposed relationship between pentagonal numbers and equal partitions? But it'll take me a few weeks to get enough understanding built up to see where it's going or if it ever actually makes it there.

****This is almost certainly written out wrong, since again I'm on mobile and can't reference any notes and my memory is shit.

 

[Jokodo]

...

Of course, odd*odd=odd. The exposition makes clear that's what I was using.

Well, it's not a link but it does provide the actual discussion I was looking for in clear terms, so thank you.

At some point I looked it up myself (the origin of the row being that I couldn't because I was on mobile and very busy and would be for much of the day), and asked the question "well, what's the deal with K, how does it grow?"

I interested to see that when I looked up the progression of K that they were in fact a progression of pentagonal numbers.

My real question is what is it about k=22,... That makes these pentagonal numbers lie about whether their result in k^2/24+1/24=**** that makes this number nonprime?

I'm going to look into the relationship of these "liars" and why they happen later.

I also found a paper from 2019 about a proposed relationship between pentagonal numbers and equal partitions? But it'll take me a few weeks to get enough understanding built up to see where it's going or if it ever actually makes it there.

****This is almost certainly written out wrong, since again I'm on mobile and can't reference any notes and my memory is shit.

My first proof is literally Adam's slightly reworded...

 

[Jarhyn]

...

Of course, odd*odd=odd. The exposition makes clear that's what I was using.

Well, it's not a link but it does provide the actual discussion I was looking for in clear terms, so thank you.

At some point I looked it up myself (the origin of the row being that I couldn't because I was on mobile and very busy and would be for much of the day), and asked the question "well, what's the deal with K, how does it grow?"

I interested to see that when I looked up the progression of K that they were in fact a progression of pentagonal numbers.

My real question is what is it about k=22,... That makes these pentagonal numbers lie about whether their result in k^2/24+1/24=**** that makes this number nonprime?

I'm going to look into the relationship of these "liars" and why they happen later.

I also found a paper from 2019 about a proposed relationship between pentagonal numbers and equal partitions? But it'll take me a few weeks to get enough understanding built up to see where it's going or if it ever actually makes it there.

****This is almost certainly written out wrong, since again I'm on mobile and can't reference any notes and my memory is shit.

My first proof is literally Adam's slightly reworded...

With discussion, and some other posts besides, and enough attention to grammar to make me confident that what I was reading was not somehow incomplete.

The second post you posted was more what I was looking for as an extension from 6k+/-1.

As to the pentagonal nature of K among the primes, I don't suppose you could connect me with some information about results such as k=51 which result in a root of 25?

But then how do the pentagonal numbers interrelate with the first two primes, 2 and 3? It is this just the fact that the first pentagon is partitioned exactly to the primes 2 and 3?

 

[lpetrich]

 Pentagonal number - A000326 - OEIS - Pentagonal numbers: a(n) = n*(3*n-1)/2. - 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 315

I don't see what they have to do with primes, because only one of them is a prime: 5.

 

[Jarhyn]

...

....

...
My real question is what is it about k=22,... That ..

...

Also lol I'm dumb, it's 51, not 22

 

[Jarhyn]

 Pentagonal number - A000326 - OEIS - Pentagonal numbers: a(n) = n*(3*n-1)/2. - 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 315

I don't see what they have to do with primes, because only one of them is a prime: 5.

Look at the factors K in p^2=24k+1 and compare the list to the series of pentagonals.

My first question, the very first on seeing Adam's post was "how does K evolve on p" and I got the first few numbers and googled them. It steps off at 25^2.

Edit: it's just that some numbers are liars about whether they lead to a prime square or not. Maybe the squares of prime squares and their squares? Again, I'll probably look closer when I get home.

 

[lpetrich]

So 24*pntg(k)+1 is the square of a prime? Where pntg(k) = k*(3k-1)/2

In general, 24*pntg(k) + 1 = (6k-1)²

So the aforementioned condition is equivalent to 6k-1 being a prime.

• 6k-1: 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335, 341, 347, 353, 359, 365, 371, 377, 383, 389, 395, 401, 407, 413, 419, 425, 431, 437, 443, 449, 455, 461, 467, 473, 479, 485, 491, 497, 503, 509, 515, 521, 527, 533, 539, 545, 551, 557, 563, 569, 575, 581, 587, 593, 599
• 6k+1: 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325, 331, 337, 343, 349, 355, 361, 367, 373, 379, 385, 391, 397, 403, 409, 415, 421, 427, 433, 439, 445, 451, 457, 463, 469, 475, 481, 487, 493, 499, 505, 511, 517, 523, 529, 535, 541, 547, 553, 559, 565, 571, 577, 583, 589, 595, 601

where I've bolded the prime numbers in the lists.

 

[Jarhyn]

So 24*pntg(k)+1 is the square of a prime? Where pntg(k) = k*(3k-1)/2

In general, 24*pntg(k) + 1 = (6k-1)²

So the aforementioned condition is equivalent to 6k-1 being a prime.

• 6k-1: 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335, 341, 347, 353, 359, 365, 371, 377, 383, 389, 395, 401, 407, 413, 419, 425, 431, 437, 443, 449, 455, 461, 467, 473, 479, 485, 491, 497, 503, 509, 515, 521, 527, 533, 539, 545, 551, 557, 563, 569, 575, 581, 587, 593, 599
• 6k+1: 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325, 331, 337, 343, 349, 355, 361, 367, 373, 379, 385, 391, 397, 403, 409, 415, 421, 427, 433, 439, 445, 451, 457, 463, 469, 475, 481, 487, 493, 499, 505, 511, 517, 523, 529, 535, 541, 547, 553, 559, 565, 571, 577, 583, 589, 595, 601

where I've bolded the prime numbers in the lists.

I can see the errors evolving as interactions od the primes not a factor of 24.

I wonder if there is a way to evolve the restriction of K to select out multiples of 5, as a specific subset of the pentagonals.

Edit: oh, I see it now: (6k-1)^2 isn't going to incorporate the factors outside of it's primorial component 6. You need to use a bigger primorial somehow, there, to weed out higher nonprimes at the expense of missing lower ones.

 

[lpetrich]

Try k = 5k' + {0, 1, 2, 3, 4} -- one gets 30k' + {-1, 1, 7, 11, 13, 17, 19, 23}

One can normalize to get within range 0 to 30: {1, 7, 11, 13, 17, 19, 23, 29}

For 6k +-1, one gets {1, 5}

For 2k + 1, one gets {1}

If one tries this approach, one will get composite numbers divisible by 7, like 49. So one will have to repeat this construction with 7: 7*(30k+{}) ...

After that, 11, 13, 17, 19, ad infinitum.