I think both will fall sooner than most other conjectures if math has a say.

math.stackexchange.com/questions/3377955/… for one of my recent collatz answers for example. This shows that not only can higher sequences be modelled with lower sequences, but any non-trivial cycle has a minimum member that is $4t+3$

pari gp gives a remainder of 7525

7525 for 9091, 3947 for 22277

The hooking parts of Collatz, link to Goldbach in the sense that if the even number before the hook is p+q with a certain property, we can show the even number afterwards will have a partition.

My guess is that neither Goldbach nor Collatz will be solved in near futute, the chances are somewhat better for the collatz conjecture.

We know almost nothing when it comes to concrete primes. We only can estimate the number of primes in a huge range. For example , we do not know whether n²+1 is infinite many often when n runs over the integers. If any pattern would help to solve Goldbach, almost surely it would have been found already. Collatz however does not deal with primes, nevertheless the dynamic structure of the collatz conjecture leaves not much hope for a proof either.

I'll be back in about 10 minutes to give a simple example or two or many.

lets say we have a number of form $2048*x+7$ , if $x$ were even, we could write it with a higher power of two coefficient and half of $x$ . It's important to note, that parity of such a number falls to the constant term. The reason this is important , is that Collatz output depends on parity. Multiplying by 3 and adding one, multiplies both terms by 3, and adds 1 to the constant term ( just like the constant terms Collatz sequence would). dividing by two eithe

either ends up at an odd linear coefficient, or divides both terms by 2. This means as long as the linear parts coefficient has suffient factors of two to continue we follow the collatz path of the first remainder term , in terms of up and down motion.

so take $x=1$ we get $2048+7=2055$, whiich goes to $6144+22=6166$, leading on again, we get 3083 , 9250,4625, 13876,6938,3469,10408,5204,2602,1301,3904,1952,976,488,244 which are all of the forms predicted algebraically.

with algebra you can give me any $x$ and I can be confident $2048*x+7$ gets you down to $243*x+1$

with Goldbach linked in with examples of $10=3+7\implies 16=5+11$

my(a=randomprime(10^18));print(2048*a+7","243*a+1) plug away.

@RoddyMacPhee hi. Sorry for interrupting.

don't mind, I was basically waiting for someone to respond anyways.

Can you answer my one question.

is it what the **** are you talking about ?

Are you referring to me.

you're the one I'm talking to, am I not ?

Sorry, here is the question.

We have a function which is F:[0, 1]>[0,1]

$x^2+(f(x))^2< 1.or equal to.

And area of f(x) from 0 to 1 =pi/4

I need to find integration from 0 to 1 $f(x) /1-x^2$

@RoddyMacPhee

okay so your first inequality is a sector of 90 degrees

OK.

@RoddyMacPhee

at least in positive coordinates. in all coordinates it's related to a circle. anyways not related easily to Collatz or Goldbach as this room was originally.

OK.

@Peter why'd you change title.