prime number patterns

Love this thread and will return in more detail.

I am prime tragic, spending more than a few thousand hours being obsessed by them. I do know there are patterns in the primes and think I have some unique discoveries to share with the world when I get enough time to formalise them - I am not a mathematician so this is very difficult.

The method I use to analyse the primes however is a little different to most. The first thing I would say is the one pattern you all must recognise is the one that results from the definition of primes, that they are only divisible by one and themselves - perhaps you think I am telling you the obvious but most people overlook this and search for patterns that result from the primes, patterns which are doomed to end or have exceptions because primes are an infinitely small subset of the all integers which are an infinitely big set. And since the primes are infinite there will be many opportunities for patterns to occur that are artifacts and not a fact of the primes. Hint - Look at how patterns form not what patterns you see.

My own theory proves a number of the postulates, most of which have already been proven by other means and some of which have not, but I need to demonstrate this more thoroughly. This is because my theory seems obviously extensible to infinity through all primes however I need to run very big numbers to prove it and make sure there is not something I overlooked. My theory is mature enough that I expect an exception to disprove it would occur fairly quickly and thus damage if not destroy my theory but I am hopeful - with good reason I think this will not be the case.

I have not found a mathematician yet who would co-operate with me who would not be concerned I am a nut, and I have not found one I can trust. Mostly due to insufficient time trying to identify someone.

TonyM

Thanks for sharing Tony, when you say trust, are you concerned that they will steal your work and claim it as their own? What are you trying to achieve now, writing a computer program to test your theories?

I suppose the Trust is just not just for the ideas but that they will be respectful to a non professional. I am quite prepared to accept criticism but not dismissal.

It is not even that, I am very busy with much in my life and whilst I am prepared to put in more time on this fanciful idea, that "I may have something new to add", I don't want to waste my time. A sentiment no doubt any collaborator will share.

My Programming skills are strong but limited and it is too time consuming for someone with the less than ideal skills. Especially when I want to do non standard manipulations on big numbers that don't fit into standard binary numbers.

I must say (sorry without references) some of the posts in this thread are "right on" with my view of the primes.

Regards
TonyM

Here's a pattern for all prime numbers greater than 2. It's my hypothesis but I've not set about a proof.

All prime numbers greater than 2 make an incomplete quadrangle.

In other words, if you try to form a matrix to represent a prime number the matrix will always be "missing the corner"

For example, three can be represented as

XX
X

and four (not prime)

XX
XX

The three represented above is missing a value in the lower right corner.

Eight (not prime) can be formed with a matrix in a variety of ways

For example

XXXX
XXXX

or

XX
XX
XX
XX

Which makes a complete matrix or "quadralangle"

But seven , which is prime, can never make a complete matrix or "quadrangle" no matter how you try to form the matrix

i.e.

XXXX
XXX

or

XX
XX
XX
X

Thus my conjecture is for all prime numbers, no matter how large, will never make a complete quadrangle.

I don't know if there is prexisting or current work that already presented this perspective, or if this is just a trivial property of prime numbers, but I've never encountered it in any text book, and who know, this might be a very first. And just to think if it is it was made by an AC on GLP!

hahahahaha

Oh, BTW, 9's is not prime (as you have cited above). It can make a complete quadrangel

i.e.

XXX
XXX
XXX

"A number greater than 1 that cannot be represented as a rectangle in this way is called a prime number." [link to www.mayhematics.com]

I think you would enjoy reading 'Uncle Petros and Goldbach's Conjecture' [link to www.amazon.com]

After two, no Prime is an even number. On the surface this would suggest no rectangle can be formed after 2 by primes.


Quick Conjecture
The Square, Cube of primes etc also form rectangles but are not prime. The need not be even as well eg; 9

Fact I have established.
If progressively considering the primality from 1,2... the first possible prime (known to that point) to be excluded from primality is the Square of the current prime.

Given every non prime must be the product of two (earlier) primes then the product will always be even and capable of being a rectangle as at least it can be two x any number.

I'm working through an example of your description for clarity on my part:

"progressively considering the primality from 1,2..."
1, 2 (prime), 3 (prime), 4 , 5 (prime) 6, 7 (prime)
I'm stopping here at 7, which we will call our "current prime".

Then you say "the first possible prime (known to that point) to be excluded from primality is the Square of the current prime"
Which would be 7x7=49, so we know 49 fails the primality test.
I am slightly confused why you say it is the first, I could argue that the first to be excluded would be the lowest product of two primes a and b (a can also equal b) beyond our current prime, in this case 3x3=9. Unless you have a list of operations that you perform in order, to exclude values from primality.




I'm confused here, 15 is non prime and is the product of 3 and 5, yet it is not even and is a rectangle.

Hope you can clear up my confusion, thanks.

expand your 1000 5 theory to 1200 everything is I. Twelves not tens help me help you.

base twelve or multiples of twelve?

Discovering connections: The prime numbers and the Fibonacci numbers.

[link to ripon.edu]

remember the scene from LOTR ?
before Gundolf died...

the place looked just like that

Are you talking about when he fell down that chasm?

[link to en.wikipedia.org]

for mathtards

If you had already excluded all the multiples of 2, all the multiples of three and 5 then when you arrive at 7 you already know all numbers are prime or not prime up to 48. 49 looks like a prime because it has not been excluded yet, it is not prime because as you discover it is 7x7

Using this method you find that the next possible prime to be excluded will all ways be the next P^2

People often understand this in reverse. That is the quickest basic way (not withstanding more advanced methods) to determine if something is prime is to attempt to divide it with each of the prime numbers first n/2 then n/3 n/5 n/7 etc... (no need to use the non primes) the moment you get an integer that divides your number you discover it is non prime if you never get a number that divides it, then it is a prime!

So when can you determine you will never find a divisor ?
In order of most basic
*If your divisor (The number you are dividing with is greater than the number you are testing. Duh

*If your divisor is greater than half....the number you are testing. well yes because the first prime is 2

*If your divisor is greater than the Square Root of the number you are testing. Because of my original observation that the next number to be excluded is N^2 - as the Square Root is the "reverse" of the square.

Regards Tony

Sieve of Eratosthenes

Make a list of all the integers less than or equal to n (and greater than one).
Strike out the multiples of all primes less than or equal to the square root of n.
Then the numbers that are left are the primes.