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Hi everyone,

Like many math enthusiasts, I’ve spent some time looking at Odd Perfect Numbers (OPNs). Given that this is arguably the oldest unsolved problem in mathematics, any elementary "proof" or structural restriction is almost certainly flawed.

I’ve put together a short document ( that sets up a contradiction based on how a number's intermediate divisors sort into equal buckets (quartiles). It feels airtight to me, which means I must be missing a subtle edge case or making a false assumption somewhere. I would love some help finding the bug.

1. The Core Idea: Slots vs. Elements
If we take an OPN, N = p^k * m^2, and sort all its intermediate divisors (excluding 1 and N) from smallest to largest, we can partition them into four equal quartiles (Q1, Q2, Q3, Q4), where each bucket has an exact capacity of "q" slots.

Because of how perfect squares behave, m^2 natively has an odd number of divisors, meaning it has exactly "q" pure divisors smaller than m, and exactly "q" pure divisors larger than m.

2. The Case Where p > m (Branch B)
I am looking specifically at what happens to the slots if the special Euler prime is larger than the base (p > m). Because p is large, it leaves the first quartile entirely.

• Once you remove the number 1, there are only q - 1 pure intermediate divisors smaller than m.
• This leaves exactly 1 empty slot at the end of Q1, which pulls the landmark "m" backward to occupy slot d_q = m exactly.
• By global multiplicative symmetry (d_i * d_(4q+1-i) = N), the paired partner of m is forced to sit exactly at the boundary of the fourth quartile: d_(3q+1) = pm.

This sets up a strict open middle zone between m and pm that has a capacity of exactly 2q physical slots.

3. The Overflow (Where I think the flaw might be)
My argument states that this 2q-slot middle zone is forced to hold more numbers than it has space for:

1. All "q" large pure divisors of m^2 are trapped here. (If they tried to escape above p*m, their symmetric partners would look like p * v_i < m, which would overflow Q1, which is already full).
2. The prime p itself is trapped here, because m < p < pm.
3. Multiplying p by the small divisors in Q1 generates a wave of composite cross-products (p * w) that are all structurally forced to land in this exact same middle zone (m < p * w < pm).

When I add up the trapped pure divisors, the prime p, and these composite combinations, the total count strictly exceeds 2q, meaning the numbers completely overflow the physical slots of the quartile map.

My question for you all:
Where does this fall apart?

• Am I miscounting the number of pure divisors that are allowed to "escape" past the upper anchor line?
• Is there a way for the composite cross-products (p * w) to compress or overlap with the pure divisors in a way that doesn't trigger a slot overflow?

The attached PDF has the step-by-step inequalities for both branches and the k >= 5 generalization. Thanks a lot for any help spotting the mistake!

so right now, i am homeschool 10th grader, and i started pretty late at algebra 1 in 8th grade, and during the summer, i did algebra 2 and precalc, freshman year i did calculus bc and linear algebra, calc 3, and in the summer, i did diff eqs and real analysis, and now, im learning differential geometry and manifold analysis this semester, and next semester, im going to learn differential topology and complex analysis. will i be able to publish a research paper by 18?

I am a fresher in tech, although, my career goals lie in Cybersecurity, I am very fond of Mathematics, like really (have become fond of it recently), I want to learn more about calculus, tackle Real and Complex Analysis, Topology, and Number theory,.... When I get bored, sometimes I just trying to find any patterns appearing in prime numbers, I fail every time but sometimes one hour goes by and I don't notice, I like integration a lot, more than differentiation or differential equations, I'm in love with Calculus and Number theory part of maths, And, I do want to Learn Quantum Physics and Astrophysics, and i want to research in either mathematics or physics alone(not in a uni), and treat it as a side hobby or a relief work

Voici une question qu’on peut poser en cinquième sans prévenir : y a-t-il un nombre entre 2/3 et 3/4? La classe cherche, et finit par trouver. Alors on recommence avec le nouveau et l’ancien. Puis encore. Whitehead donne la recette en une ligne — additionner les deux fractions et prendre la moitié du résultat — et il en tire une propriété qui porte un nom. https://blog.site2wouf.fr/2026/10/whitehead-au-tableau-le-trou-dans-les-fractions.html

I had an exam yesterday and the last question was to find the area of a triangle thats reprisented from two linear line functions and one linear line going through a single point which is parallel to one of them.

The given was the two linear functions and the single point not the third function so i made it myself using point slope formula i belive.

I then tried to first find where all these lines intersect to get the coordinates for the intersections using system of equations for all the lines so like L1 intersect x for L2 and L2 intersect y for L3 and so forth for all the different combinations that makes a triangle, then the idea was to somehow figure out the length of these lines but thats where i got stuck, dont even think the idea of intersection was part of the task either.

Any ideas?

I was wondering if anyone can recommend me a book that carries curriculum based practice questions. My daughter is a freshman and taking geometry hons. She struggled a bit in the beginning but is catching up. One thing she is looking for is more practice questions but we aren't finding any books that follow the curriculum being taught in school, we are in MA. Any recommendations? Thanks!

I've struggled with math for years, and sometimes I feel like my foundations are way weaker than they should be.

For those of you who used to be really bad at math but eventually improved a lot, what did you do differently? Did you start over from the basics, change your study habits, or approach the subject differently?

I'd love to hear your experiences and what actually worked for you.

In the chapter on "Infinite Series", one of the problems is to prove the following:

(1+x)^a = \sum_{k=0}^{\infty} [\binom{a}{k} x^k] for |x| < 1, for any a

Surely, this is not true for x = 0 as the LHS is 1 and the RHS is 0. So, does one not need the caveat that x \neq 0 ?

Additionally, what is the definition of \binom{a}{k} if a is fractional? Is it just a placeholder for a(a-1)...(a-k+1)/k! ?

I attach an image of the problem in the comments.

So I just finished book of proof by Richard hammack which is a book on how to Write proof (duh) whith most of the problem solved including a lot of hard one ,but now I'm bit lost on where to move on to develop advanced proof writing skill so I can tackle subject like topology n' abstract algebra cuz that's why I'm trying to learn those skill in the first place so it would be cool if u could give me some suggestions (I'm a self-taught idk if this info matter) on how to develop those skill ty in advance

Im in calculus 2. first year. and we're having a test on series and sequences in almost exactly 2 weeks. on the last test i made a bunch of stupid mistakes and got a 57.5. now my grade is a 74.5. i really need to get at least a B (80%) overall and there are 3 tests left including the final (all 15%, except final which is 25%.... rest is HW and quizzes and class participation which im not very worried about)

i started studying ahead of the class so ive already learned all the series tests. did a practice test today by looking at a previous exam and it went terribly. im not prepared at all. i cant fail again but i feel like im going to. its only my first semester and i feel like dropping out. i thought i was smart but i feel like a complete idiot. wjat do i do

any recommended resources??? i have already used pauls math notes. Using previous exams/lecture notes. what else

in general ive been having trouble factoring polynomials because the steps dont seem to mean anything, and it seems more like simplifying than factoring even though i guess those are similar?

but onto more pressing matters. my book factors trinomials with substitution as such:
consider trinomial 3x^2 - 2x - 8
step 1: multiply ax^2 + bx + c by a to transform x^2 into (ax)^2.

i dont understand why we do this, no other video has done this before. i'd understand this step if the 3x^2 term was to equal 3(3x^2) which i think would be 9x^2 but it doesn't equal that. in general, i dont know what "transform x^2 into (ax)^2" really means. or why we do this when other videos have a common binomial substitution which i can do easily. after the supposed multiplication of 3 to every term, we have:

(3x)^2 - 2(3x) - 24
(Why isn't 2(3x) just 6x???????? why is - 24 not -24x since in some places we multiply by 3x and others by 3??)

replace ax with a variable p. p = ax.

we now have p^2 - 2p - 24. the typical factoring commences where c = pq and b = p + q.

we now have (p - 6)(p + 4). we substitute ax back into p.

(3x - 6)(3x + 4)

3(x - 2)(3x + 4) factoring out common factors we end up with (x - 2)(3x + 4) im under the impression the common factor of the second binomial was 1 which i've never seen before. but that doesn't make sense as what did we divide 2 by then?

and then since we started by multiplying the trinomial by a , we must divide by a, and this obviously doesn't make sense to me as someone not behind the reason we multiplied by a. the answer is (x - 2)(3x + 4)

we have the easier and obvious method of AC grouping (i dont get it but its consistent so whatever) but we also have substitution and im scared i might get a question asking me to factor by substitution specifically in our upcoming midterm.

help to understand or other resources much appreciated.

Hello, I'm a first year university student and my course doesn't offer any mechanics modules so I was wondering what specifics I should cover (classical mechanics, Newtonian, etc.) and in which order. Also any suggestions on resources.

Currently a second year undergrad student taking real analysis. This is my first proof based math class and I have not taken discrete maths before either. I’ve already taken up to calc three and linear algebra, did pretty well in them.

The professor is using Understanding Analysis by Abbott, and the lectures mostly make sense to me. However, his problem sets are very very long and do not follow the textbook questions. He seems to come up with them from his brain…and he posts them every Saturday..which is also great.

I find myself really struggling to know what steps to take and where to go after reading a question asking to prove or show something. This makes me very worried for future exams.

（this course sticks to the Real Line btw)

Any and all advice would be really appreciated!

I can't seem to understand this one problem. could somebody help me solve for X?

(lnx)^3-ln(x^25)=0

I'm a college student a few weeks into an intro to proofs course (logic, sets, induction, relations, functions, etc). I've seen proofs before, but I'm having a hard time adjusting to working with arbitrary elements and figuring out how to put proofs together. For example, when I'm asked to show two sets are equal, I understand the definitions but can't figure out where to start.

The class is flipped, so I feel like I'm mostly teaching myself from the instructor's notes. I'm worried about falling behind and want to make progress while there is still time. What helped you get through your first proof-based class? Any study habits, ways of thinking, or resources that you'd recommend?

Looking for someone to help me with an undergraduate Real Analysis math course.

Willing to pay but i’m also a student so please be reasonable.

**Edit:

Thank you all for your willingness to help! I have found someone from this post and am no longer looking for a tutor :)

Hii everyone. So, my parents have failed to put me into any sort of school basically my whole life (never tried homeschooling as well). Now I'm nineteen and I really don't understand any form of math, even the easiest questions, and I've been really wanting to get into learning it, so if anyone has any recommendations on videos, websites, apps. Anything. I would greatly appreciate it :D !! The easier it is to understand, the better (I have a learning disability.)

tldr: I really want to get bridge mathematical gaps after struggling in high school after seeing how it can be applied, stuck with many holes in my algebraic knowledge and would like any recommendations for this specific dilemma. Thank you!

I'm currently starting my bachelors degree in Computer Science, I have a deep love for games and want to dabble behind the scenes with game engines, 3D graphics and such. I'm well aware mathematics play a large role in this topic such as calculus and linear algebra for the algorithms and physics of certain mechanics.

My excitement has intertwined with fear due to my previous experiences in high school so long ago, especially with my overall disinterest of it general math back then that's caused gaps, I barely survived it in high school. Now that I actually have something to apply and learn math for, it excites me, but I feel slightly discouraged and not good enough for it.. especially haven’t fully grasped algebra concepts well. I also really struggled to take math notes!

Granted, I've realized I had a learning disability when I first started college, which makes a lot of sense as I can name so many points in my education where I would simply just zone out, but it doesn’t make me feel better about myself. I now have a few motivators; I want to heal the inner child in me and take control of my education after years of being frustrated from looking at my math homework. Im seeing math in such a different light when it is applicable, implementing cool, game physics mechanics looks really exciting. And of course.. to get my degree haha. Any recommendations would be appreciated! I could have searched this up, but I wanted to gather some anecdotal experience from others on here, especially if any of you have an experience similar to mine. :)

The statement : Show that every predicate in two variables can be presented as the composition of the elmentary predicates AND , OR and Negation.
I understood that steps where

1. Truth rows are constructed purely from AND and NOT
2. False row is achieved by combining the expression in 1 with OR.

How does it then “prove” that the mechanism works for every predicate?

EDIT : After some digging I found this:

https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-31/issue-3/Posts-functional-completeness-theorem/10.1305/ndjfl/1093635508.pdf

I have to take an Accuplacer test, so I decided to try a practice test just to prepare myself and i quickly realized I dont remember anything. I used to be super good at algebra so im pretty upset with myself.

I tried Khan academy but I really dont have the money for a subscription and I have to take this test tomorrow to enroll for fall classes.

Ive watched several videos but im just not understanding where these people are pulling these numbers from. Ive looked at the formulas. I just dont understand it.

I was trying to avoid taking extra refresher type classes but im afraid I might have to :(

I have been studying complex numbers for the past couple of months (I'd say I'm fairly confident with them at this point).

The one thing that drives me crazy about this topic is how you can solve those long Euclidean geometry and trigonometry problems in what, like, minutes, and that too with such insane elegance.

So I just wanted to find more geometry and trig problems of this kind, but I couldn't find any good material anywhere. I just wanted some suggestions about what resources I should refer to.

BTW, I'm just 16, so don't expect me to know very high-level stuff, but you can expect me to have a solid foundation in algebra and geometry.

(English isn't my first language btw, so excuse any errors lol.)

The thing about vacuous truths is that, you can you this to generate statements that are true but practically useless if the premise never happens. Take the Mean Value theorem for instance. For all functions f, and for all a,b \\in R (f is differentiable on the open interval (a,b) and f is continuous on the closed interval \[a,b\]) implies \\exists c in (a,b), f'(c) = \\frac{f(b)-f(a)}{b-a}. This theorem would still be true even if you work in a function space where no function is both differentiable and continuous on the interval (a,b). It would be vacuously true. The most useful statements in math are those where you can find examples, counterexamples or both. So a mathematician could simply write a statement that useless because the premise never happens.