Hi Everyone, I recently posted in this reddit about my proof for a Twin Prime Conjecture.

In that thread I got feedback that my Theorem 3 and Lemma 5 were broken and I had gaps in my logic.

I believe I have addressed those issues now, and so if you are willing I am grateful if you can have a second look.

I am looking for people to show me where the proof is currently weak as they see it.

Because, I think I have it, and it's rigorous, and if I don't have it, I think the remaining gaps in logic are small now if there are any at all.

I do intend over the next week to attempt to put this into Lean and try to verify it that way.
But in the meantime, I am here to ask for some potential roadblocks any of you might see for me doing that, so I can catch them earlier, because I am unfamiliar with Lean.

This new paper fixes the problems brought up about my previous Theorem 3 and Lemma 5.

It is the same mathematical construction, however I have approached the logical statements in a different method than I previously did, and I believe I have overcome the hurdles now.

This paper also has a companion python function which works as a finite representation of how the function works.

The python function itself cannot act as the proof because you would have to run the function for an infinite length of time with infinite computer resources in order to use that as proof.
However it does serve as an easy to understand implementation of the function and proof posed in the paper.

A quick read of the Abstract can give you a good understanding of exactly how I am intending to solve the TPC

Thank you in advance for anyone willing to spend some time on this to help me out.
Also I am hosting the paper on my own site incase Zenodo is down.
https://prosperousplanet.ca/research/TwinPrimes.pdf

I was having a discussion with an enemy about how Wonder Woman 1984 was a government plot to cover up 1984 to stop any ideas of that sort of thing and then he said that if you add up all the numbers it makes 2000. Says I, that’s 22. But that got me wondering is there any number that doesn’t end in 0 that if you multiply by another number you get 0s before the ending digit. Something like 12*x=102
Ĕ̈d̆̈ĭ̈t̆̈:̆̈ S̆̈ĭ̈n̆̈c̆̈ĕ̈ Ĭ̈ c̆̈l̆̈ĕ̈ă̈r̆̈l̆̈y̆̈ w̆̈ă̈s̆̈n̆̈t̆̈ c̆̈l̆̈ĕ̈ă̈r̆̈ ĕ̈n̆̈ŏ̈ŭ̈ğ̈h̆̈,̆̈ h̆̈ĕ̈r̆̈ĕ̈ ă̈r̆̈ĕ̈ s̆̈ŏ̈m̆̈ĕ̈ m̆̈ŏ̈r̆̈ĕ̈ ĕ̈x̆̈ă̈m̆̈p̆̈l̆̈ĕ̈s̆̈
2̆̈3̆̈1̆̈x̆̈(̆̈y̆̈)̆̈=̆̈2̆̈3̆̈0̆̈1̆̈
1̆̈1̆̈1̆̈1̆̈x̆̈(̆̈y̆̈)̆̈=̆̈1̆̈1̆̈1̆̈0̆̈1̆̈
T̆̈h̆̈ĕ̈ s̆̈ă̈m̆̈ĕ̈ n̆̈ŭ̈m̆̈b̆̈ĕ̈r̆̈ b̆̈ŭ̈t̆̈ ŏ̈n̆̈l̆̈y̆̈ ă̈ 0̆̈ b̆̈ĕ̈f̆̈ŏ̈r̆̈ĕ̈ t̆̈h̆̈ĕ̈ l̆̈ă̈s̆̈t̆̈ d̆̈ĭ̈ğ̈ĭ̈t̆̈

This is a question about whether there is a function that would show all 3 of these given changes in variables. I mean the shortest function possible (even if its necessarily large).

I’m not the brightest with probability problems and I’ve been doing trial and error on my strategy on how to deal with them. I’m in college and some colleagues suggested that I just have to do a lot of practice, which is true but it doesn’t work as well for me compared to studying other math topics. The word play really gets to me leaving me confused and stuck until I just stare at the question. I try to create visual models for every problem I answer but it just fuels my confusion, probably because it doesnt make sense. Sure I get some questions right, here and there which I’ve encountered before, but when I see fresh unfamiliar exam questions, I go blank even though I’ve studied the topics for it. Are there any ways, decision paths, mind maps, structured framework, or approach you guys follow that works well for you whenever you get stuck on a probability problem?

Set Operations:

This shirt makes ZERO freaking sense. Basically set of all elements that aren't in b or c and thats what I freaking did. Istg math is so freaking stupid.

According to the table, can you infer about f(x)?

The options were lim_{x→2} f(x) = 1, lim_{x→2} f(x) = ∞, lim_{x→10} f(x) = ∞, lim_{x→10} f(x) = 2.

Obviously, the given table matches the function f(x)=0.75-0.05*log10(|2-x|). One of my friends had the same thought process, so they picked the second choice. I knew that the teacher wanted us to choose the first one, so I did.

According to the teacher, the correct answer was the first one, but I feel that it isn't fair that my friend didn't get points. We tried to argue, but she wouldn't give the points.

Is it my friend's problem or the teacher's problem that he lost points on that problem?

I am not a true mathematician, so don't be too hard on me for my confusion here.

https://www.youtube.com/watch?v=J72AKZtUpgY

I just watched the first 25 minutes of this video to see how exactly this stuff is taught. It is defined that differentials are changes along a tangent line from a point on a curve. Then it is derived that the slope at the point f'(x) is equal to the ratio of the differentials dy/dx.

Question 1: If we remind ourselves that dy/dx as a ratio of differentials is not the definition of the derivative, but is rather "derived" to be equal, is there really a problem here? I always hear it is non-rigorous, but particularly when looking at it in this way I don't see a problem yet.

Question 2: The only issue is then An issue I see is dy/dx is being used as notation for the ratio of two finite numbers, and simultaneously as the limit of a ratio (how derivative is defined in real analysis). So we have f'(x) (derivative) = dy/dx (derivative) = dy/dx (ratio of two numbers). Now this is problematic. But to me it seems more like an abuse of notation than lacking rigor.

Can we really hammer down where the non-rigor lies? I'd really appreciate if possible people could respond in line to the two questions. I am comfortable with real analysis perspectives, where differentials are never mentioned. But I look back on my calculus days, and have a hard time pinpointing the "problem"

I need help with this problem i need to find the area of the big box, the AIs seem to break when i post it and try to give me the answer 1^2 +3^=area of big box=10cm

Which doesn't make sense please i need a clear explanation

Exericse 11.4.10: Use the definition of logarithm to show that (a) log_b(b^x) = x for every real number x, (b) b^{log_b(x)} = x for every positive real number x

Property 7.2.5:

Solution for (a):

Let log_b(b^x) = y
Then b^y = b^x               by definition of logarithm
Then y = x                   by property 7.2.5
Since log_b(b^x) = y and y = x, log_b(b^x) = x, for every x in R
QED

Solution for (b):

Let log_b(x) = y
Then b^y = x                by definition of logarithm
Also, b^{log_b(x)} = b^y    by substitution
Since b^y = x and b^{log_b(x)} = b^y, b^{log_b(x)} = x, for every x in R^+
QED

Is my solution correct?

I’ve been thinking. Yk how like after like 7 moves in chess there are 6 sextillion chess positions. So it’s likely that it’s a position never seen before (excluding common long openings) How different is that from words? Like after twelve words is it Likely this exact sentence been said?

Tried doing this problem but it doesn’t make sense to me. I’ve tried making the bigger rectangle equal 1000 but that’s not right. Don’t know what it means by what does it equal.

I understand that, for k ≠ 0, lim x-->0 (sinkx)/x is an indeterminate form.
Thus we have to use the fundamental trigonometric limit (i.e. lim x-->0 sinx/x = 1).

Then, my professor asked "What if k = 0?". He said the solution is to use direct substitution and NOT to use the fundamental trigonometric limit. Why is this?

I asked my professor about this and he said it is because sin(0) divided by anything is 0. And a limit is not about finding the functional value??? I don't really understand what he meant. Please help me out.

also, the precise definition of limit, the epsilon delta thingy is out of scope for my course. So, please let me know if I need to learn about that to fully understand what my professor meant. (I'm studying engineering)

I understand that they pulled the 1/5 out of the integral, and I understand that they chose -(x/5) for their u and differentiated it which gives you du = -1/5dx. Next you need to get dx by itself so that you can substitute dx for du and have the whole integral in terms of one variable u right? So to get dx by itself you would multiply both sides by -5 right? Which would give you -5du = dx. You would then swap the dx in the integral for -5du and if you pull the -5 out of the integral as well it get's multiplied by the 1/5 that was already there. -5 times 1/5 = -1, not 25. What am I missing here?

I just can't... I don't understand it... No ELI5 helped me yet.

Here's an example with a brief explanation of what I know and where I get stuck.

Let R = Z[x] and I the set of all the polynomials from R whose coefficients add up to 0. Prove that I is the ideal of R. Is this ideal also the principal ideal? Prove that R / I is isomorphic to Z.

So, Z[x] is a set of polynomials with integers. I appears to be a specific subset of Z[x] and therefore should be a subset of R. By definition, a ring is an Abelian group for addition and a semigroup for multiplication. Next, a subring must have closure, addition/subtraction test and a multiplication of 2 elements within itself also stays inside.

An ideal further expands on the idea of a subring and only asks for 'external' multiplication by taking an element from the ring, multiplying it with an element from the subring and checking if the product remains in the subring.

Principal ideal is a specific ideal in which only 1 element can be used for multiplication and it generates the entire set. The common simple example of it I came up with is (Z/nZ), integers modulo n.

I would say I understand theory well enough (except the R / I which is the quotient ring concept I am not entirely familiar with) but I simply can't understand this or any similar task. I don't know how to solve it and literally need baby steps.

In the task itself I understand that coefficients adding up to 0 would probably mean that the 'free part' is always the answer so x - 1 would be a simple example of the kind of polynomial the ring is describing. How do prove the I is the ideal of R? I need to show it's a subring first, but I don't know how to do that either and even following the definition feels impossible to me.

Okay, let's take some 3x - 3 and 2x - 2 as examples. (3x - 3) - (2x - 2) = x - 1, so it checks out I guess. Next, multiplying with an element from the ring. Since this rule only applies to the subring I, I can take some x (the simplest polynomial from R) and then (3x - 3)x = 3x^2 - 3x, also checks out so is the ideal here done with?

Now, how would I show it's a principal ideal? Maybe it's generated by x - 1? You can add/subtract it as many times as you want and you'll get all the elements. The proof regarding the quotient ring is completely unknown to me and I would appreciate if someone could explain. Just so you guys know I had no solution to this task and was genuinely lost, now when writing it I came up with some examples which could demonstrate some things and my brain started working, hopefully it's all good.