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(07/04/2022, 07:30 PM)tommy1729 Wrote: i was not able to find nonzero numbers w such that w^2 = 0.
I assume no such nonzero w exist. Epsilon, from the dual numbers when squared equals zero. And epsilon is not zero.
Please remember to stay hydrated.
ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
Posts: 1,924
Threads: 415
Joined: Feb 2009
I want to add that in most cases " the candidates " are multiplication tables where the base elements satisfy most of the time the equation :
a*(b*c) = (a*b)*(a*c) up to sign ( +1 or -1 factors )
so we kinda get latin squares with the property of " pseudo self-associative " or " pseudo self-distributive ".
So it is very much connected to virtual knots and latin quandles.
notice how self-associative ( or self-distributive although the other term is more standard ) implies non-associative for a latin square !
The number of latin quandles of size n times n with respect to the factorization of n is therefore very interesting too.
(the factorization of n is like a subgroup/subalgebra thing therefore the research )
Although this clarifies alot , there remain many questions.
iterating self-associative operators also comes to mind, this is also a non common subject just like tetration.
This is post nr 1729 of tommy1729 , hurrah !!!
regards
tommy1729
Tom Marcel Raes
" truth is what does not go away when you stop believing in it. "
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
ok.
I need to add some important stuff.
there have basically been 3 strategies , and until now I mentioned only 2 :
The techniques are for multiplication of 2 distinct units in the multiplication table.
The techniques are basically equations satisfied most of the time.
All of them require commutative. And we want latin squares.
And non-associative.
Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.
However we do use for the units :
(x y)^2 = x^2 y^2 = +/- 1.
The strategies are based on working without +/- signs first and then adding them later.
1) the first strategy is to use the quandle equations :
a * (b * c) = (a * b) * (a * c)
a * b = b * a
a * a = a
and at the end (when the table is complete , but before adding the signs +/- ) replace a * a with +/-1 instead of a.
I posted a pic of this before here in dimension 5.
But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )
2)
" Anti-associative "
a * (b * c) = - (a * b) * c
a * b = b * a
An example was given with dim 8 ; I posted a pic of that before.
3)
The mersenne structure.
We want the property A ( for the units at least )
x * (y * 1/y) = x = (x * y) * 1/y
so ignoring signs this implies and not counting the real part as dimension here :
in dimension 3 :
a b = c
a c = b
b c = a
Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.
The number of dimensions or equiv the number of unit variables is then restricted.
Lets see
if we have x elements and add one element Y,
then we must multiply every one of the x elements with x , add the x elements and the one Y element.
So we end up mirroring the x elements with Y and adding the one Y.
In other words :
adding a single element and still requiring the " triangle structure " or equivalently property A , we must have
2 x + 1 elements.
so the function 2x + 1 is a way to increase the dimension.
Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.
example
a b = c
a c = b
b c = a
leads to
x_1 = (a,a)
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)
etc
so we get x_9 elements and the products are similarly defined
examples :
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)
So we end up with 2 function to construct higher dimensions :
2x + 1
and products
xy
.
IF we start with 1 or 3 and only use iterations of 2x + 1 we get
1,3,7,15,...
those are the mersenne numbers : 2^n - 1.
and when they are mersenne primes they have no substructure.
But we also had the products method.
so for instance dimension 21 is possible :
(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.
It is a fun question to study what numbers can be reached and similar number theory and constructions.
( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )
Ok so without the reals we can have 21 dimensions.
adding the reals we get 22 dimensions.
However 22 / 4 is not integer so it is not solvable.
That is the basic idea.
The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 ) is then dimension 8.
Im investigating it.
3*3*3 + 1 = 28
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )
Yes strategy 3 is my new favorite.
Im not sure if not nilpotent can be defended though.
Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.
***
remarks
I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.
more precisely , apart from this deep algebra ,
the equation
f(n+1) = a f(n) + b
with a and b relatively prime
has no solutions of type 2^n + 1 or similar.
but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...
***
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
also for fans of number theory :
A debunked conjecture is this :
Let the prime p be a mersenne exponent : 2^p - 1 is prime.
Call the set of those p : E.
Then the smallest primitive root g_p (mod p) is always of the form
g_p = 2^A * P
where A is an integer and P is in the set E.
example
min primitive root ( 2^132049 - 1 ) = 26 = 2 * 13 = 2^1 * (mersenne exponent =13)
However a counterexample is
min primitive root ( 2^42643801 - 1 ) = 11.
11 is a prime and not an element of E !!
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
02/10/2023, 11:53 PM
(This post was last modified: 02/12/2023, 02:00 PM by tommy1729.)
(02/05/2023, 10:40 PM)tommy1729 Wrote: ok.
I need to add some important stuff.
there have basically been 3 strategies , and until now I mentioned only 2 :
The techniques are for multiplication of 2 distinct units in the multiplication table.
The techniques are basically equations satisfied most of the time.
All of them require commutative. And we want latin squares.
And non-associative.
Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.
However we do use for the units :
(x y)^2 = x^2 y^2 = +/- 1.
The strategies are based on working without +/- signs first and then adding them later.
1) the first strategy is to use the quandle equations :
a * (b * c) = (a * b) * (a * c)
a * b = b * a
a * a = a
and at the end (when the table is complete , but before adding the signs +/- ) replace a * a with +/-1 instead of a.
I posted a pic of this before here in dimension 5.
But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )
2)
" Anti-associative "
a * (b * c) = - (a * b) * c
a * b = b * a
An example was given with dim 8 ; I posted a pic of that before.
3)
The mersenne structure.
We want the property A ( for the units at least )
x * (y * 1/y) = x = (x * y) * 1/y
so ignoring signs this implies and not counting the real part as dimension here :
in dimension 3 :
a b = c
a c = b
b c = a
Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.
The number of dimensions or equiv the number of unit variables is then restricted.
Lets see
if we have x elements and add one element Y,
then we must multiply every one of the x elements with x , add the x elements and the one Y element.
So we end up mirroring the x elements with Y and adding the one Y.
In other words :
adding a single element and still requiring the " triangle structure " or equivalently property A , we must have
2 x + 1 elements.
so the function 2x + 1 is a way to increase the dimension.
Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.
example
a b = c
a c = b
b c = a
leads to
x_1 = (a,a)
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)
etc
so we get x_9 elements and the products are similarly defined
examples :
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)
So we end up with 2 function to construct higher dimensions :
2x + 1
and products
xy
.
IF we start with 1 or 3 and only use iterations of 2x + 1 we get
1,3,7,15,...
those are the mersenne numbers : 2^n - 1.
and when they are mersenne primes they have no substructure.
But we also had the products method.
so for instance dimension 21 is possible :
(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.
It is a fun question to study what numbers can be reached and similar number theory and constructions.
( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )
Ok so without the reals we can have 21 dimensions.
adding the reals we get 22 dimensions.
However 22 / 4 is not integer so it is not solvable.
That is the basic idea.
The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 ) is then dimension 8.
Im investigating it.
3*3*3 + 1 = 28
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )
Yes strategy 3 is my new favorite.
Im not sure if not nilpotent can be defended though.
Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.
***
remarks
I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.
more precisely , apart from this deep algebra ,
the equation
f(n+1) = a f(n) + b
with a and b relatively prime
has no solutions of type 2^n + 1 or similar.
but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...
***
regards
tommy1729
So our dimensions without the real dimensions are
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
It is an interesting integer sequence.
Density theorems would be nice.
but that is a bit off topic.
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
(02/10/2023, 11:53 PM)tommy1729 Wrote: (02/05/2023, 10:40 PM)tommy1729 Wrote: ok.
I need to add some important stuff.
there have basically been 3 strategies , and until now I mentioned only 2 :
The techniques are for multiplication of 2 distinct units in the multiplication table.
The techniques are basically equations satisfied most of the time.
All of them require commutative. And we want latin squares.
And non-associative.
Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.
However we do use for the units :
(x y)^2 = x^2 y^2 = +/- 1.
The strategies are based on working without +/- signs first and then adding them later.
1) the first strategy is to use the quandle equations :
a * (b * c) = (a * b) * (a * c)
a * b = b * a
a * a = a
and at the end (when the table is complete , but before adding the signs +/- ) replace a * a with +/-1 instead of a.
I posted a pic of this before here in dimension 5.
But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )
2)
" Anti-associative "
a * (b * c) = - (a * b) * c
a * b = b * a
An example was given with dim 8 ; I posted a pic of that before.
3)
The mersenne structure.
We want the property A ( for the units at least )
x * (y * 1/y) = x = (x * y) * 1/y
so ignoring signs this implies and not counting the real part as dimension here :
in dimension 3 :
a b = c
a c = b
b c = a
Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.
The number of dimensions or equiv the number of unit variables is then restricted.
Lets see
if we have x elements and add one element Y,
then we must multiply every one of the x elements with x , add the x elements and the one Y element.
So we end up mirroring the x elements with Y and adding the one Y.
In other words :
adding a single element and still requiring the " triangle structure " or equivalently property A , we must have
2 x + 1 elements.
so the function 2x + 1 is a way to increase the dimension.
Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.
example
a b = c
a c = b
b c = a
leads to
x_1 = (a,a)
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)
etc
so we get x_9 elements and the products are similarly defined
examples :
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)
So we end up with 2 function to construct higher dimensions :
2x + 1
and products
xy
.
IF we start with 1 or 3 and only use iterations of 2x + 1 we get
1,3,7,15,...
those are the mersenne numbers : 2^n - 1.
and when they are mersenne primes they have no substructure.
But we also had the products method.
so for instance dimension 21 is possible :
(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.
It is a fun question to study what numbers can be reached and similar number theory and constructions.
( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )
Ok so without the reals we can have 21 dimensions.
adding the reals we get 22 dimensions.
However 22 / 4 is not integer so it is not solvable.
That is the basic idea.
The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 ) is then dimension 8.
Im investigating it.
3*3*3 + 1 = 28
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )
Yes strategy 3 is my new favorite.
Im not sure if not nilpotent can be defended though.
Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.
***
remarks
I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.
more precisely , apart from this deep algebra ,
the equation
f(n+1) = a f(n) + b
with a and b relatively prime
has no solutions of type 2^n + 1 or similar.
but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...
***
regards
tommy1729
So our dimensions without the real dimensions are
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
It is an interesting integer sequence.
Density theorems would be nice.
but that is a bit off topic.
regards
tommy1729
My friend mick posted that density question on mathstackexchange and it got some upvotes and interest.
https://math.stackexchange.com/questions...ne-numbers
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
(02/10/2023, 11:53 PM)tommy1729 Wrote: (02/05/2023, 10:40 PM)tommy1729 Wrote: ok.
I need to add some important stuff.
there have basically been 3 strategies , and until now I mentioned only 2 :
The techniques are for multiplication of 2 distinct units in the multiplication table.
The techniques are basically equations satisfied most of the time.
All of them require commutative. And we want latin squares.
And non-associative.
Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.
However we do use for the units :
(x y)^2 = x^2 y^2 = +/- 1.
The strategies are based on working without +/- signs first and then adding them later.
1) the first strategy is to use the quandle equations :
a * (b * c) = (a * b) * (a * c)
a * b = b * a
a * a = a
and at the end (when the table is complete , but before adding the signs +/- ) replace a * a with +/-1 instead of a.
I posted a pic of this before here in dimension 5.
But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )
2)
" Anti-associative "
a * (b * c) = - (a * b) * c
a * b = b * a
An example was given with dim 8 ; I posted a pic of that before.
3)
The mersenne structure.
We want the property A ( for the units at least )
x * (y * 1/y) = x = (x * y) * 1/y
so ignoring signs this implies and not counting the real part as dimension here :
in dimension 3 :
a b = c
a c = b
b c = a
Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.
The number of dimensions or equiv the number of unit variables is then restricted.
Lets see
if we have x elements and add one element Y,
then we must multiply every one of the x elements with x , add the x elements and the one Y element.
So we end up mirroring the x elements with Y and adding the one Y.
In other words :
adding a single element and still requiring the " triangle structure " or equivalently property A , we must have
2 x + 1 elements.
so the function 2x + 1 is a way to increase the dimension.
Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.
example
a b = c
a c = b
b c = a
leads to
x_1 = (a,a)
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)
etc
so we get x_9 elements and the products are similarly defined
examples :
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)
So we end up with 2 function to construct higher dimensions :
2x + 1
and products
xy
.
IF we start with 1 or 3 and only use iterations of 2x + 1 we get
1,3,7,15,...
those are the mersenne numbers : 2^n - 1.
and when they are mersenne primes they have no substructure.
But we also had the products method.
so for instance dimension 21 is possible :
(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.
It is a fun question to study what numbers can be reached and similar number theory and constructions.
( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )
Ok so without the reals we can have 21 dimensions.
adding the reals we get 22 dimensions.
However 22 / 4 is not integer so it is not solvable.
That is the basic idea.
The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 ) is then dimension 8.
Im investigating it.
3*3*3 + 1 = 28
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )
Yes strategy 3 is my new favorite.
Im not sure if not nilpotent can be defended though.
Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.
***
remarks
I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.
more precisely , apart from this deep algebra ,
the equation
f(n+1) = a f(n) + b
with a and b relatively prime
has no solutions of type 2^n + 1 or similar.
but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...
***
regards
tommy1729
So our dimensions without the real dimensions are
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
It is an interesting integer sequence.
Density theorems would be nice.
but that is a bit off topic.
regards
tommy1729
More number theory :
consider the sequence b(n) generated by
a) 1 is in the list
b) if x is in the list , 3x - 1 is in the list
c) if x,y are in the list then x y is in the list
It seems this sequence grows about twice as slow :
if a(n) is the sequence of generalized mersenne numbers ( or whatever they should be named ) :
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
then it seems b(2n) is close to a(n).
b(22) = 50
a(22) = 99
a(11) = 43
b(11) = 22
b(37) = 95
b(n) is
1,2,4,5,8,10,11,14,16,20,22,23,25,28,29,32,40,41,44,46,47,50,55,56,58,59,64,65,68,70,74,80,83,86,88,92,95,...
and it shares many analogue properties of a(n).
I remember I was only 13 when I considered these sequences.
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
(02/14/2023, 11:30 PM)tommy1729 Wrote: (02/10/2023, 11:53 PM)tommy1729 Wrote: (02/05/2023, 10:40 PM)tommy1729 Wrote: ok.
I need to add some important stuff.
there have basically been 3 strategies , and until now I mentioned only 2 :
The techniques are for multiplication of 2 distinct units in the multiplication table.
The techniques are basically equations satisfied most of the time.
All of them require commutative. And we want latin squares.
And non-associative.
Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.
However we do use for the units :
(x y)^2 = x^2 y^2 = +/- 1.
The strategies are based on working without +/- signs first and then adding them later.
1) the first strategy is to use the quandle equations :
a * (b * c) = (a * b) * (a * c)
a * b = b * a
a * a = a
and at the end (when the table is complete , but before adding the signs +/- ) replace a * a with +/-1 instead of a.
I posted a pic of this before here in dimension 5.
But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )
2)
" Anti-associative "
a * (b * c) = - (a * b) * c
a * b = b * a
An example was given with dim 8 ; I posted a pic of that before.
3)
The mersenne structure.
We want the property A ( for the units at least )
x * (y * 1/y) = x = (x * y) * 1/y
so ignoring signs this implies and not counting the real part as dimension here :
in dimension 3 :
a b = c
a c = b
b c = a
Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.
The number of dimensions or equiv the number of unit variables is then restricted.
Lets see
if we have x elements and add one element Y,
then we must multiply every one of the x elements with x , add the x elements and the one Y element.
So we end up mirroring the x elements with Y and adding the one Y.
In other words :
adding a single element and still requiring the " triangle structure " or equivalently property A , we must have
2 x + 1 elements.
so the function 2x + 1 is a way to increase the dimension.
Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.
example
a b = c
a c = b
b c = a
leads to
x_1 = (a,a)
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)
etc
so we get x_9 elements and the products are similarly defined
examples :
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)
So we end up with 2 function to construct higher dimensions :
2x + 1
and products
xy
.
IF we start with 1 or 3 and only use iterations of 2x + 1 we get
1,3,7,15,...
those are the mersenne numbers : 2^n - 1.
and when they are mersenne primes they have no substructure.
But we also had the products method.
so for instance dimension 21 is possible :
(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.
It is a fun question to study what numbers can be reached and similar number theory and constructions.
( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )
Ok so without the reals we can have 21 dimensions.
adding the reals we get 22 dimensions.
However 22 / 4 is not integer so it is not solvable.
That is the basic idea.
The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 ) is then dimension 8.
Im investigating it.
3*3*3 + 1 = 28
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )
Yes strategy 3 is my new favorite.
Im not sure if not nilpotent can be defended though.
Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.
***
remarks
I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.
more precisely , apart from this deep algebra ,
the equation
f(n+1) = a f(n) + b
with a and b relatively prime
has no solutions of type 2^n + 1 or similar.
but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...
***
regards
tommy1729
So our dimensions without the real dimensions are
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
It is an interesting integer sequence.
Density theorems would be nice.
but that is a bit off topic.
regards
tommy1729
More number theory :
consider the sequence b(n) generated by
a) 1 is in the list
b) if x is in the list , 3x - 1 is in the list
c) if x,y are in the list then x y is in the list
It seems this sequence grows about twice as slow :
if a(n) is the sequence of generalized mersenne numbers ( or whatever they should be named ) :
1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...
then it seems b(2n) is close to a(n).
b(22) = 50
a(22) = 99
a(11) = 43
b(11) = 22
b(37) = 95
b(n) is
1,2,4,5,8,10,11,14,16,20,22,23,25,28,29,32,40,41,44,46,47,50,55,56,58,59,64,65,68,70,74,80,83,86,88,92,95,...
and it shares many analogue properties of a(n).
I remember I was only 13 when I considered these sequences.
regards
tommy1729
The sequence generated by 2x and 3x-1 but not xy is known :
https://oeis.org/A190807
and similar but not identical.
25 is not in A190807.
regards
tommy1729
Posts: 1,924
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I might use this idea from gottried.
https://math.eretrandre.org/tetrationfor...hp?tid=730
Well I had some extension ideas for it.
But it is complicated.
regards
tommy1729
Posts: 1,924
Threads: 415
Joined: Feb 2009
https://math.stackexchange.com/questions...ne-numbers
For the extended mersenne numbers that are so cruxial here, I am looking for special cases :
4 conditions for X :
**
X = an extended mersenne number
**
X = a product of two extended mersenne numbers A and B, where
A is a product of 2 primes C,D where C and D are extended mersenne numbers
and B is a prime.
Or equivalent X = BCD where B,C,D are primes and extended mersenne numbers.
**
X is squarefree.
**
(X + 1)/4 is a prime P and P is NOT an extended mersenne number.
OR simplified :
3 conditions for X :
X = BCD where B,C,D are primes and extended mersenne numbers.
**
X is squarefree.
**
(X + 1)/4 is a prime P and P is NOT an extended mersenne number.
( notice primes of the form 4n + 1 are not extended mersenne numbers )
regards
tommy1729
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