Don’t mistake meek for weak
or mistake silent for quiet
or mistake my intent for luck
Cuz, I’ll give a whole new meaning
to the term mindfuck
and I’m just gettin’ warmed up
ask La Columbiana Guapa, without any drama
or those black cats I’ll call ’em two goats
then ask Cuddles what she knows
Ya, I’ll drop the “tiger” if you want it
It hogs a lot of material, that 36 grit
And about that photo the rock was in
you’re right, Ms. Wife, without the knife, they could’ve been twins
you’ll be findin’ that with the Queen of Diamonds
There’s a dime, a King, the Jack of Hearts and me
I’m Big A, and you’ll see we’re rollin’ that BDE
And cousin, now that you know it’s true
Do ya think the world is ready for 1-oh-2?
For everyone that screams about proof
Ladies, just sing about the Groove
if that’s not enough for you fools
Ms. Goings you didn’t get played
just tell Bert that’s what he gets for drinking the kool-aid
if you wanna call me out, don’t give it much thought
if you wanna call me out, call me Prof
Remember, not everyone who stumbles and falls
are forever lost
If that’s too much or too funky
Pronounce it with an Umpty
And if you wanna know who I am and what I’ll be,
C’mon y’all, say it with me
because with great power, there must always be
Great responsibility
and a fuckin’ Jedi Knight like me.
My Father loved my mother dearly
cautiously showing us how to live fearlessly
and the daily livin’
of my mother’s vision
a promised life together, a Family
My brother was my mother’s first Greatest Love
her first taste of being a mother
he was and is a trailblazer
A great but reluctant leader
She always said she re-raised her big brother
My sister was my mother’s Pride
a little Queen with a big fire inside
she would be so proud to see
she raised the woman she wanted to be
A great mother who will fight or die by your side
I was my mother’s Hope, her Little Prince yet to be
despite the hurt, what I learned, changed that, see
Her hoped-for future
Imperfectly sutured
Now, there’s something else I’ve become
I am my mother’s Vengeance
and I am her greatest weapon
“What would my mother say?” Really? Okay…
You obviously don’t know who you’re talking about
She would say, “You run it up, you leave no doubt.”
“You make them remember the day they fucked with, Andrew Steven Bounds.”
“With your shield, or on it, son,”
“you go in it to win it, or you won’t come out”
“Now calm down and show them what you’re about”
Thanks for reminding me what she’d wanna see
So, are we gonna kick ass or chew bubblegum?
‘Cause I’m fresh out of one…
ON THE DECIDABILITY OF THE COLLATZ CONJECTURE: A
SORTING ALGORITHM FOR (2,4) MOD 6. THE COLLATZ
CONJECTURE SOLUTION
ANDREW S. BOUNDS
Part 1. INTRODUCTION
Part 2. DOMAINS
The Domain of the seed values, x, are the integers, and the range are the natural
numbers.
Given: 3n + 1 = y1 where: if n = 2×1, y
2 , if n = 2×2 + 1, then 3n+1
2 = y2
This domain definition yields three trivial loops, two loops at x1 = 0, and another
at x1 = 1 for y1 = 2x as it then feeds into the odd trivial loop of x2 = 0, which
gives x1 = 2 is equivalent to x2 = 0. It also yields a single domain for all x.
Every power of two (2a) will collapse to the trivial odd loop if we divide by 2,
so we need to ensure that our algorithm can cycle through all possible evens that
can have powers of two, divide by two if it is a power of two, and kick out of our
”slot machine” if it is not.
We need to account for every even and odd input, and every possible output.
Our function cannot have solutions that repeat, but the more values it covers, the
more likely we are to find a power of two.
Ideally, we would also be able to extend the approach to the domain of negative
integers. If there is a linear solution, or set of solutions, then we should be able
to extend those to the negative integer domain while maintaining the same range,
just negative.
Part 3. COPRIME/MOD 6/BASE 6 NOTATION
With the realization that only powers of two will collapse to the trivial loop of
x=0 for all n = 2x and is the only point in which they crossover, we can use the
expanded equation of the case where n = 2x + 1:
(1) 3(2x + 1) + 1 = 6x + 4
Received by the editors 2024.
2020 Mathematics Subject Classification. Number Theory.
©XXXX American Mathematical Society
1
2 HAILSTONE PROBLEM
0 mod 6 1 mod 6 2 mod 6 3 mod 6 4 mod 6 5 mod 6
2(3x) 2(3x+1) 2(3x+2)
6x+0 6x+1 6x+2 6x+3 6x+4 6x+5
3(2x) 3(2x)+1 3(2x)+2 3(2x+1) 3(2x+1)+1 3(2x+1)+2
Table 1. Positive coprime expansion
x 0 1 2 3 4 5 6 7
3(2x+1)+1
21 ) mod 6 2 5 2 5 2 5 2 5
3(2x+1)+1
22 mod 6 1 4 1 4
Table 2. Oscillating values
to begin understanding the possible combinations. This would lead to the use of
a 6-coprime expansion, also a C mod 6 notation. This also leads to the expansion
of all 6-coprimes listed in the diagram (Table 1). This coprime expansion will also
be the basis for expanding to the negative integers. As this expansion is inclusive
of all the coprime expansions of 2 and 3, all possibilities for the Collatz expansion
are represented.
It is a nominally trivial notion that this can also be expanded to work with any
linear expressions with odd coefficients and constants less than the coefficient simply
by doubling the lead coefficient to determine the different factors when dividing by
Example: Ax + c where A is an odd integer and c is a number where 0 ≤ c < A
would give a working value of (2A)x + (2c) for all divisions of 2 by also including
the odd expressions of x.
Part 4. Coprime Expansion
As illustrated in the above diagram (Table 1), the odd and even coprime multiples
of 3 and their related doublets as even numbers are completely mapped. This
represents every possible permutation of co-prime linear equations for 3’s. This
also gives a structure for separating out that which can be a power of two and that
which cannot. Although this does not limit every possibility, it significantly limits
the number of possibilities that can be an output. This also has an impact on the
sequencing of every possibility when dividing, by leaving off the multiples of 3 and
their doubles.
With the above table (Table 1), it is trivial to see that the odd and even outputs
of the algorithm using 3n + 1 with n = 2x + 1 or n = 2x have distinct values. What
is not so trivial to see is that the function 3n + 2 is the other half of the solutions.
The coprime or modular form also allows us to see a recurring pattern in the
seed values for x. As seen in Table 2.
This also gives a maximum slope of 3(2x + 1) + 2 or 6x + 5 for any iteration
path that would climb toward infinity. This would also require that every seed
value produce an odd value for the seed value x and there is only one division by
2 every iteration. The distribution of evens and odds in the seed value x supports
all previous work that took the probability as abstract. This applies to all previous
AMS ARTICLE TEMPLATE 3
2 coprime 2(3x) 2(3x-1) 2(3x-2)
6 coprime 6x-0 6x-1 6x-2 6x-3 6x-4 6x-5
3 coprime 3(2x) 3(2x)-1 3(2x)-2 3(2x-1) 3(2x-1)-1 3(2x-1)-2
Table 3. The proper negation of the Collatz function
work on 3n + 1 and 3n + 2 on the Collatz Conjecture as these are the only outputs,
and therefore secondary seed values, for the Collatz Conjecture.
Part 5. Negative Integer Domain
If we change our domain, we need to continue the negation across all steps in
the expression, so all terms are negated in turn after x:
(2) 2x + 1 = 2x − 1 = n, 3n + 1 = 3n − 1 = 3(2x − 1) − 1 = 6x − 4
This gives us an expansion of the expression similar to the positive integers:
This form also allows us to see a particular pattern (Table 3).
This allows us to see the direct expansion to negative integers and that every
iteration following the form outlined above will give the same result for the domain
of negative integers. All of the above shows that the previous findings of other
researchers and in this paper apply to the negative integers when using this form.
Part 6. Path Between Iterations
(3) 3(2x + 1) + 1
2p where p = 2a and also 3(2x + 1) + 1
2q where q = 2a + 1
which yields:
(4) 3(2x + 1) + 1
2p = 3(2x + 1) + 1
2q as the only path between iterations.
Part 7. Conclusion
With the above information, we can determine, for any given x:
(1) The only outputs possible from any given input are divisions by 2a and the
parity of a tells you if it is 6x + 1 or 6x + 5 and anything else results in
another division by 2.
(2) The only way for the process not to stop and reset is to find a power of 2.
(3) The even numbers corresponding to 6x + 2 and 6x + 4 can be sorted by this
process into 3 distinct outputs, the odd roots, q = 2a + 1, not a power of
two correspond to 6x+5, and the even roots, p = 2a, which are not a power
of 2 correspond to 6x + 1. The powers of two would be the third output.
The x = 0 being the only crossover point with an odd and an even power
of two. All other values are not an output of the described algorithm.
4 HAILSTONE PROBLEM
(4) If we change the domain of x to negative integers, then all terms and cor-
responding operations must be properly negated to show the same results
for that domain, but the algorithm can be completely mapped to the set of
negative integers.
(5) The only path that produces any loop is the one that is x = 0, which yields
an even and odd parity in the same co-prime set (2 and 4).
With the above points, we can determine that the 6-coprime expansion encompasses
all the behaviors and examples of the algorithm output possibilities. The expansion
also shows differentiation by the parity of the input value of x and by the number
of divisions by 2, or 2a. The parity of a is also determinant to which output is
achieved, pairing even values of a with even values of x. The only other outputs
available are the divisions by 2, which yield the trivial loop when there are two
multiples of 2 in the same set of 6.
Abstract. ABSTRACT: In this it is shown that there is an algorithm for
sorting (2,4) mod 6 (a.k.a. Syracuse problem, Collatz conjecture, Hailstone
problem, etc.) is true through the use of 6 coprime notation and seeing the
problem as an algorithm for sorting roots of even numbers to find powers of
two from integers-reduced-to-odd inputs, a robust algorithm in fact. Previous
work focused on the 3n+1 structure, but this approach shows that it only
encompassed half of the solution. 3n+2 being the other half.
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I am so happy you stayed interested
And of course, apology accepted
And my equal reply
In full, in kind
Now let's see if I can't make good the investment
I'm going to give this a helluva try
cause if I'm even mostly right
There's lots to do
more than me and you
and I'm a fucking Jedi Knight!
The gap after the song was indeed a dramatic pause
I wanted to be clear, plus it's some of the best advice I ever got
When he was singin'
Keep their heads ringin'
And made a phone call, like it or not
Now, A phone call to you, is a party line to me
They wouldn't have believed, and they needed to see
A tail wag to you
Hi, I'm Andrew
and it doesn't get any more real than me
And now, Experiment successful! let's say twice
the third time was to show you what it was like
Mrs. look-you-in-the-eyes-when-I-do-it
He thought He could do it
Was the fourth time because of the fight?
And you already know I am not a witch
You just can't stop scratching that itch
Stop being so bold
You know you've been told
Now watch those stumbles Mitch
And I'm not trying to take down the man
It's more like Denny's, a real Country Slam
and Brother, t'was more than little chuckle
When I got to call ya all Hat and no Buckle
It really was good to finally see your fam
So many out there that I could mention
The little Tumblers attention
And a blush
for my Comic Crush
And the White Queen up in her mansion
This isn't a joke, it's gravely important
I love to laugh, but y'alls perspective is distorted
If this is real
do you really feel
That your satisfaction is to what it should be devoted?
Please.
For the dude whose head looks like a cue
And Dave the 8-ball that was with you
"Hi guys", you heard that too right?
Yeah, you heard right that night
And, you doubting bitches, I did too
Wait, Hold up there buddy, It smells a little Musky
Now I'm pretty sure that bet was for 2 big hunks of money
dunno if it was a Mil,
or was it Bill?
But that Saturday Night Live shit was funny
'And to take out the cast'
they say the best for last 😉
Moses Supposes His Toeses Are Roses Song
'bows'
but always, always, always, make them laugh…
oh yeah, I had to look back but it made me smile
You guys were the reason that she had to lie
So, thanks for the shower, Big Willie Style…
A wonderful human piece from Dr. Jen Gunter. Go check out her blog for a treasure trove of good information, heartfelt advice, and fantastic writings.
This is from another blog, go check it out. They’re looking for contributors if you’re a writer…
http://www.unchainedjane.com/2016/05/dont-be-shithead-be-accountable.html?m=1
Coq means rooster in French. But it has been named after the name of its inventor, Thierry Coquand. It is an interactive theorem prover. It is not an automated theorem prover but includes automatic…
Source: Coq proved four colour theorem
andrewsaysblog 