Prime numbers

[sunshaker]

I have little math, But i came across this fact about prime numbers which i thought was interesting and worth a share, Most properly old to those with math,

23X23=529 =(24X22)+1

 

29X29=841=(24X35)+1

 

31X31=961=(24X40)+1

 

37X37=1369=(24X57)+1

 

41X41=1681=(24X70)+1

I was wondering if there is any sequence to this? Between the multiples of 24?

 

A little find like this to me, brings a new interest and lets me explore where i would otherwise perhaps never go.

So my first thought was 3x24 -1=73, So i googled 73 ,

 

73 is an emirp, meaning that the reverse of 73, that is, 37, is also a prime number. Interestingly, 73 is also the 21st prime number while 37 is the 12th prime

 

So i now know about "emirp" numbers, I know this is basic stuff, But we all learn something new everyday.

Edited June 2, 2014 by sunshaker

[imatfaal]

the 24 thing is quite simple

 

p is a prime

 

p^2 -1 = p^2-1^2 = (p+1)(p-1)

 

of the three consecutive numbers p-1,p,p+1 we can say p must be odd (it's a larger than 2 prime) therefore p-1 and p+1 must be even

 

both p-1 and p+1 must be divisible by two (they're even) and one of two consecutive even number must be divisible by four.

 

in every three consecutive numbers one of them must be divisible by three - it cannot be p (it is prime) therefore either p-1 or p+1 is divisible by three

 

so (p-1)(p+1) is divisible by 2 AND 4 AND 3 - which is also known as 24

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on the intervals; no I believe not. This would lead to an almost complete knowledge of primes - and we do not have that. AS shown above the 24 thing is pretty simple and adds no extra knowledge or information to the problem

 

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emirp and permutable primes (ie 337, 373, 733) are interesting on a certain level - but tied to the number system and I think get boring quickly.

[Acme]

Just to throw a Prime trivia factoid into the mix, the product of most Primes and a Perfect number is Abundant by twice the Perfect.

 

For example, multiply the Perfect number 28 by the Prime 5. 5x28=140 Factors of 140: {1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140} Sum the factors excepting 140 itself: 1+2+4+5+7+10+14+20+28+35+70=196 Then 196-140=56 and 56=2*28.

 

There are exceptions, such as 7x28=196. Find the factors of 196: {1, 2, 4, 7, 14, 28, 49, 98, 196} Sum those factors, excepting 196 itself: 1+2+4+7+14+28+49+98=203 Then 203-196=7 and 7≠2*28

 

24 is a different kind of exception here because it is Abundant by twice the Perfect 6, but it is not the product of a Perfect and Prime. Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24} Summing: 1+2+3+4+6+8+12=36 and 36-24=12 and 12=2*6.

 

So it goes.

[sunshaker]

Thank you for explanations, Something else I have learnt today, (including abundant and perfect numbers).

I have been looking for something I could think on for a while, Perhaps I will look for that "complete knowledge of primes".

"fools rush in" .

[Acme]

Thank you for explanations, Something else I have learnt today, (including abundant and perfect numbers).

I have been looking for something I could think on for a while, Perhaps I will look for that "complete knowledge of primes".

"fools rush in" .

You're welcome.

 

Here's a pictorial/graphic arrangement of Primes that may convey the nature of the morass you are planning to jump into.

 

 

Were there such a thing as complete knowledge of Primes, then this pattern would not be random. This link gives the construction details as well as several other spiral-themed graphs of Primes. >> Ulam spiral @ Wikipedia

[sunshaker]

You're welcome.

 

Here's a pictorial/graphic arrangement of Primes that may convey the nature of the morass you are planning to jump into.

 

 

Were there such a thing as complete knowledge of Primes, then this pattern would not be random. This link gives the construction details as well as several other spiral-themed graphs of Primes. >> Ulam spiral @ Wikipedia

 

Never knew there where so many ways to view primes(interesting), I noticed different prime triangles https://www.google.co.uk/search?q=prime+triangles&safe=off&es_sm=93&tbm=isch&tbo=u&source=univ&sa=X&ei=fxeNU9fWKqXn7AbplICgDA&ved=0CEYQsAQ&biw=1152&bih=723,

Had a quick go putting a couple together, Only from 1 to 105, then doubling 1 to 210, early days yet,

https://alphaomegadotme.files.wordpress.com/2012/05/prime-triangles.xls

[Acme]

Never knew there where so many ways to view primes(interesting), I noticed different prime triangles https://www.google.co.uk/search?q=prime+triangles&safe=off&es_sm=93&tbm=isch&tbo=u&source=univ&sa=X&ei=fxeNU9fWKqXn7AbplICgDA&ved=0CEYQsAQ&biw=1152&bih=723,

Had a quick go putting a couple together, Only from 1 to 105, then doubling 1 to 210, early days yet,

https://alphaomegadotme.files.wordpress.com/2012/05/prime-triangles.xls

The triangles are interesting, though unlike Ulam's spiral there are many ways to construct them. (There are 8 different orientations of Ulam's spiral, but they preserve the relative positions of the primes.)

 

Have fun and if I think of something else on the theme I'll post.

[sunshaker]

You're welcome.

 

Here's a pictorial/graphic arrangement of Primes that may convey the nature of the morass you are planning to jump into.

 

 

Were there such a thing as complete knowledge of Primes, then this pattern would not be random. This link gives the construction details as well as several other spiral-themed graphs of Primes. >> Ulam spiral @ Wikipedia

REWIND ABOVE PICTURE, YOU WILL GET TO A POINT. The further you expand to higher primes, The less ordered it becomes.

 

 

I Cannot help but think this reminds me of the expansion of universe, Starts from a point and expands outwards, Order within the chaos.

THIS MAY BE MOVED TO SPECULATION, Which i am ok with as I am speculating.

 

The "Point" in primes is one, Every number then expands out from within one as I have tried to show below.

I have used triangles joining 2 then four, for each stage.

 

 

Edited June 3, 2014 by sunshaker

[imatfaal]

Careful; you can lose yourself and your self in depictions and investigations of primes! Or of any part of number theory - as I think Acme might testify - the practitioners would claim they have found themselves of course; but whichever way it can be consuming.

 

And I think you have a glitch around 191-192 in last diagram above

[sunshaker]

Careful; you can lose yourself and your self in depictions and investigations of primes! Or of any part of number theory - as I think Acme might testify - the practitioners would claim they have found themselves of course; but whichever way it can be consuming.

 

And I think you have a glitch around 191-192 in last diagram above

Cheers just fixed "glitch".

[Acme]

REWIND ABOVE PICTURE, YOU WILL GET TO A POINT. The further you expand to higher primes, The less ordered it becomes.

 

I Cannot help but think this reminds me of the expansion of universe, Starts from a point and expands outwards, Order within the chaos.

THIS MAY BE MOVED TO SPECULATION, Which i am ok with as I am speculating.

 

The "Point" in primes is one, Every number then expands out from within one as I have tried to show below.

While there is a valid philosophical argument to considering 1 a Prime, there is a much greater set of practical arguments to not consider it so. That all numbers stem from Unity is the very basis of number theory.

 

Careful; you can lose yourself and your self in depictions and investigations of primes! Or of any part of number theory - as I think Acme might testify - the practitioners would claim they have found themselves of course; but whichever way it can be consuming. ...

As do conscientious spelunkers, I roll out a ball of twine as I go. Along the way I also keep it in my mind that I am engaged in a recreational pursuit that is as much -if not more- about the journey than about the destination.

[sunshaker]

A question for those who know far more than me about prime numbers, I know little about primes but love patterns,

 

Just thought I would look at periodic table and look for patterns within prime elements,

 

I looked first at element 45 rhodium, To see if there was a pattern, I noticed it seemed to be a "mirror element" each element each side was the same distance apart,

example each side of 45, 61/29 67/23 71/19 73/17 this carrys on but aluminium gets in the way, Which i will look into.

 

There seems to be a few where elements each side expand at regular intervals, What i wanted to know if there is a name for this?

Not just within elements but within all prime numbers?

 

Edited June 7, 2014 by sunshaker

[Acme]

A question for those who know far more than me about prime numbers, I know little about primes but love patterns,

 

...

There seems to be a few where elements each side expand at regular intervals, What i wanted to know if there is a name for this?

Not just within elements but within all prime numbers?

...

Primes are randomly disturbed among the positive integers. All manner of alluring prime patterns rear their pretty heads, only to sooner or later disappear in the froth of random behavior and taking with them those who dared too close an approach.

 

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. ~ Leonhard Euler

[Unity+]

Primes are randomly disturbed among the positive integers. All manner of alluring prime patterns rear their pretty heads, only to sooner or later disappear in the froth of random behavior and taking with them those who dared too close an approach.

We have to establish that there is a difference between seemingly random and actually random, since I am not aware of a proof that primes are actually. "randomly" distributed.

[sunshaker]

I have included 1 as a prime, Even though it as been excluded in modern times because it is more convenient in advanced number theory( excluded by definition).

 

I have been looking at the primes within the periodic table, As i mentioned above, There seems to be a mirroring effect each side of element rhodium45,

But there are 4 elements without a mirror element,

 

These are Elements 2, 3, 5, 13, I was wondering why these elements have no mirror element,

I have a few ideas but came across fibonacci primes http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#fibprimes

 

 

The first 5 fib/primes are 2, 3, 5, 13, and 89, 89 is what my mirror primes go upto but is a mirror of 1.

 

The first 300 Fibonacci numbers http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html
n : F(n)=factorisation
0 : 0
1 : 1
2 : 1
3 : 2
4 : 3
5 : 5
6 : 8 = 23
7 : 13
8 : 21 = 3 x 7
9 : 34 = 2 x 17
10 : 55 = 5 x 11
11 : 89

I was wondering if there was anything to this fibonacci prime effect through the elements?.

 

Edited June 11, 2014 by sunshaker

[Acme]

We have to establish that there is a difference between seemingly random and actually random, since I am not aware of a proof that primes are actually. "randomly" distributed.

Just keep telling yourself that.

.

I have included 1 as a prime, Even though it as been excluded in modern times because it is more convenient in advanced number theory( excluded by definition).

Really? Can you give some non-modern example of 1 as a convenient Prime?

 

 

I was wondering if there was anything to this Fibonacci prime effect through the elements?

Yes; it's inversely proportional to the quasi-additive beta-function of the lower bound of the derivative for sunflower-seed chocolate cake brownies.

 

God may not play dice with the universe, but something strange is going on with the prime numbers. ~ Paul Erdos

[sunshaker]

 

Really? Can you give some non-modern example of 1 as a convenient Prime?

 

 

 

God may not play dice with the universe, but something strange is going on with the prime numbers. ~ Paul Erdos

 

As i said I have little math, But myself thinking 1 was a prime, Soon realized it was not classed as a prime, But since looking into primes I realized 1 was once thought of as a prime,

The last Mathematician to publically call 1 a prime was Henri lebesgue

http://planetmath.org/henrilebesgue

 

So yes 1 is not a "prime", but 1 can be thought of in many ways as I am coming to realize., 1 is a large part of what primes are, Reminds me of another thread is Pluto a planet? It used to be before those who "know best" decided it was no longer a "planet".

 

1 to a non mathematician seems a good place to start. But 1 will only play a part if it fits.

 

I was interested in understanding how the prime/factorization numbers played a part, and 1 was the mirror of 89 the last prime factorization number within p/table. And why primes mirror each side of 45 and why 1 also fits whether it is or is not classed as a prime.

[Acme]

...

So yes 1 is not a "prime", but 1 can be thought of in many ways as I am coming to realize., 1 is a large part of what primes are, Reminds me of another thread is Pluto a planet? It used to be before those who "know best" decided it was no longer a "planet".

 

1 to a non mathematician seems a good place to start. But 1 will only play a part if it fits.

 

I was interested in understanding how the prime/factorization numbers played a part, and 1 was the mirror of 89 the last prime factorization number within p/table. And why primes mirror each side of 45 and why 1 also fits whether it is or is not classed as a prime.

My concern is your getting all balled up with perceiving patterns to have some larger meaning or application. When it comes to Primes this is not an uncommon trap, but a trap it is. As to chemistry I have no idea, though I know one guy that has got all balled up with the periodic table and Pascal's triangle rather than Primes.

 

As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality. ~ Albert Einstein

[PeterJ]

I cannot understand why the prime distribution is called random. It seems the wrong use of the word. Can the output of a determinate and predictable process be called random?

 

The OP's observation that squares of primes is always at 6n+1 is the first thing that I spotted about the primes when I looked. It hardly suggests randomness. But 'randomness' will have a few definitions I suppose.

 

I would much prefer to call 1 a prime and call 2 and 3 non-prime. Then we can say that all primes occur at 6n+/-1. I have no idea of the implications of this approach. To me 0,1,2,3, are 'axiomatic' numbers, and unique as such. I'm not suggesting changing mathematics, heaven forbid, just that it may sometimes be useful to think in this way.

[imatfaal]

Peter - as we have discussed before there is no known process to generate primes - we can only look for them amongst numbers. If I give you a prime and ask you for the next ten primes you have to sieve for them - no formula in existence will allow you to short this search and produce them.

 

We know where many primes lie, and where even more primes are likely to lie - but too often we do not know if any will come in between. You know why the 6n+-1 works - it is simple arithmetic; but nothing tells us which instances will be prime or not. In effect it is no more profound than saying that all primes larger than 2 lie at 2n+1 (ie all primes larger than 2 are odd)

[PeterJ]

Nope, sorry Imatfaal, I still don't agree. There is a very easy to way to generate primes. This is to generate them the way they are generated in the first place. It's not a quick way of doing it but speed is not the point. The products of the primes are easy enough to predict, and the gaps are the primes. It's just a combination of waveforms. No randomness in sight.

 

What you say is right, but it doesn't seem to be an objection. The fact that we can predict the 6n+/- rule by simple arithmetic is surely proof that there is no randomness, just no convenient algorithm to calculate the next prime. There never will be, since the primes are not causally related. Knowing the previous one does not help us predict the next. The relationship is between their products.

 

I wonder if we're just disagreeing about the meaning of 'random'. I know it has various technical meanings.

[imatfaal]

Nope, sorry Imatfaal, I still don't agree. There is a very easy to way to generate primes. This is to generate them the way they are generated in the first place. It's not a quick way of doing it but speed is not the point. The products of the primes are easy enough to predict, and the gaps are the primes. It's just a combination of waveforms. No randomness in sight.

 

What you say is right, but it doesn't seem to be an objection. The fact that we can predict the 6n+/- rule by simple arithmetic is surely proof that there is no randomness, just no convenient algorithm to calculate the next prime. There never will be, since the primes are not causally related. Knowing the previous one does not help us predict the next. The relationship is between their products.

 

I wonder if we're just disagreeing about the meaning of 'random'. I know it has various technical meanings.

 

Not "generate" primes - but in fact "find" primes. If we could generate primes I would give you the rule, and you produce the next prime; this is impossible at present (and probably forever). What we can do at the moment is take a number and check to see if it is prime. Surely you can recognise the difference?

 

Your wave idea is nice - but it merely shows a space where primes might exist; but crucially not where they must!

 

The reason we say it is random (I am not sure if real mathematicians do so) - is that there is no rule that allows exact prediction of the next term

[PeterJ]

I don't really understand why we disagree. There is no might 'about' it. The squares of primes occur at 6n+1. Where is the randomness in this? How can this be true if they are random. If the primes were random, then there might be a prime at 5^2. But we know there is not, because the primes are predictable. They never occur where there is a product of a prime.

 

It would not be difficult to build a machine that produces the prime sequence. So in what sense are they random?

[Acme]

I don't really understand why we disagree. There is no might 'about' it. The squares of primes occur at 6n+1. Where is the randomness in this? How can this be true if they are random. If the primes were random, then there might be a prime at 5^2. But we know there is not, because the primes are predictable. They never occur where there is a product of a prime.

 

It would not be difficult to build a machine that produces the prime sequence. So in what sense are they random?

As imatfaal pointed out, Primes are random because there is no rule that produces them. For example if I ask you to give the 34556992975862856962656923^th Prime there is no [known] way to do it except to start from 2 and check every number for primality and tally found Primes 'til you find the one I asked for. As to 1 being Prime, we already covered that.

 

I don't understand why you are arguing. Mathematicians have no problem with saying Primes are randomly distributed, so your feelings about it really don't matter. If you don't want to call them random, then don't call them random.

[PeterJ]

I still don't see why we should call a distribution random just because it is difficult to recreate. If it can be recreated, then how can it be random? The primes are not like raindrops.

 

If you tell me all the primes up to P then I will calculate the next prime. I might need a few years, depending on size of P, but it can be done. The next prime does not appear randomly. It's just a bit of a pain to calculate.

 

I'm happy to change my mind, it makes no difference to me whether the primes are called random, but why should I?