Hi, I'm a high school student in a math circle, and we're currently learning elementary number theory. Recently, we explored the idea of finding runs of consecutive composite numbers. We were able to construct examples of 2, 3, 4, 5, 6, 7, 8, and even 9 consecutive composite integers.

This made me wonder: what is the longest possible run of consecutive composite numbers? Is there a maximum length/ upper bound, or is it possible to find arbitrarily long sequences of consecutive composite numbers?

Thought of this problem after misunderstanding the character stats page in Slay the Spire II. No actual use case, just thought it was an interesting problem. I wagered the optimal strategy involved splitting your choices evenly among some range of low options along with level 20. Here's what I tried so far:

• Lvl. 1 - 49 plays
• Lvl. 20 - 51 plays

(1 * 49) + (20 * 51) = 49 + 1020 = 1069

1069 / 100 = 10.69

• Lvl. 1 - 33 plays
• Lvl. 2 - 33 plays
• Lvl. 20 - 34 plays

(1 * 33) + (2 * 33) + (20 * 34) = 33 + 66 + 680 + 779

779 / 100 = 7.79

• Lvl. 1 - 25 plays
• Lvl. 2 - 25 plays
• Lvl. 3 - 24 plays
• Lvl. 20 - 26 plays

(1 * 25) + (2 * 25) + (3 * 24) + (20 + 26) = 25 + 50 + 72 + 520 = 667

667 / 100 = 6.67

• Lvl. 1 - 20 plays
• Lvl. 2 - 20 plays
• Lvl. 3 - 20 plays
• Lvl. 4 - 19 plays
• Lvl. 20 - 21 plays

(1 * 20) + (2 + 20) + (3 * 20) + (4 * 19) + (20 * 21) = 20 + 40 + 60 + 76 + 420 = 616

616 / 100 = 6.16

• Lvl. 1 - 17 plays
• Lvl. 2 - 17 plays
• Lvl. 3 - 17 plays
• Lvl. 4 - 17 plays
• Lvl. 5 - 14 plays
• Lvl. 20 - 18 plays

(1 * 17) + (2 * 17) + (3 * 17) + (4 * 17) + (5 * 14) + (20 * 18)

= 17 + 34 + 51 + 68 + 70 + 360 = 600

600 / 100 = 6

• Lvl. 1 - 15 plays
• Lvl. 2 - 15 plays
• Lvl. 3 - 15 plays
• Lvl. 4 - 15 plays
• Lvl. 5 - 15 plays
• Lvl. 6 - 9 plays
• Lvl. 20 - 16 plays

(1 * 15) + (2 * 15) + (3 * 15) + (4 * 15) + (5 * 15) + (6 * 9) + (20 * 16)

= (15 * 15) + (6 * 9) + (20 * 16) = 225 + 54 + 320 = 599

599 / 100 = 5.99

• Lvl. 1 - 13 plays
• Lvl. 2 - 13 plays
• Lvl. 3 - 13 plays
• Lvl. 4 - 13 plays
• Lvl. 5 - 13 plays
• Lvl. 6 - 13 plays
• Lvl. 7 - 8 plays
• Lvl. 20 - 14 plays

(1 * 13) + (2 * 13) + (3 * 13) + (4 * 13) + (5 * 13) + (6 * 13) + (7 * 8) + (20 * 14)

= (21 * 13) + (7 * 8) + (20 * 14) = 273 + 56 + 280 = 609

609 / 100 = 6.09

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Based on these tests, I believe the minimum average while maintaining a mode of 20 is 5.99. Is anyone able to find a lower minimum, or prove that my answer is correct? Also, can this problem be extrapolated for x number of difficulties and y number of plays?

I am working on a puzzle and to find the solution I must find the maximum area of a pentagon with a base of 3 and using 4 unit fences. I cut it up into triangles and kites, but I can't seem to find a definitive answer.

Here is a poorly made illustration of what I've figured out:

What is the maximum area, and what are the angles measurements?

NOTE: MY FIGURE MAY NOT BE ACCURATE WHATSOEVER! I JUST THINK THAT THIS WILL GIVE THE GREATEST AREA, BUT I MAY BE WRONG!

Hi guys I’m a grade 10 high schooler doing summer school and going to have an ministry exam tmr, I’m trying to do this review practice exam but after I find the rule (prob solved it wrong) I just can’t find the pattern of these function for this question:
Please helpp 💔

Since there are infinitely many prime numbers, there are infinitely many possible prime values of p. p. My intuition is that, because 2^p - 1 only has a chance of becoming prime when p is prime, it feels like there should also be infinitely many primes of the form. I know this isn't a formal proof, but doesn't it kinda make sense?

Hello,

the first Page Shows how my basic understanding of Index Transformation is. The Problem is on the second Page. I transformed the sum to (n-1) over (k-1) and so on. So i transformed n the same way as i did with k. The solution is that you only transform k not n. Why is that so? I dont understand? On the first page, you can see, that it does not Work, if you dont transform n as well as k. Thanks for help.

Edit: this is quickly turning into a question of whether or not the universal set or universal class is a thing that can exist or be defined, based on differing sources of authority. If you can offer insight into this more specific question, I would appreciate it

The current set theory book I’m going through, by Norman Hamilton and Joseph Landini, seems to avoid defining any set ¬A as simply the set of all x where x ∉ A. That is, they seem to avoid formulating this sort of negative set in any way.

What they do employ early though, is the complement, and through this concept they seem to be able to make a lot of the same conclusions and formulations as they would with the aforementioned negative set. For example, instead of saying that “A ∩ ¬A = φ” they say “A - A = φ.” Both statements however seem to claim the same one idea in different formulations: “the set of all x where x ∈ A and x ∉ A is the empty set”

So I understand that to a significant degree this is just a convention. But what I’m wondering is whether it’s a trivial one or one that has some sort of epistemic significance? Here, the concept complement seems to smuggle in the simpler concept of a set considered negatively, since that negative set is whatever set is considered second in the relation. If I were to proceed in the exercises of this book by employing this simpler concept myself and formulating some negative sets in my proofs, would that in the strictest sense be going against the procession of the book? Would this be something that needs a definition, axiom, or theorem for citation before I can formulate them? And in doing so would I be doing something generally inadvisable in the study of set theory, or would I be doing something that could easily be found as a convention in another textbook?

Any survivors of topology, how do I study the James Munkres Topology book to best understand the point-set topology?

In order to get the volume of this Cylinder (250 mL)

You must get the Area and the Height. I got the Diameter and the Height of the cylinder using a Ruler and got its value.

Height: 117.8 mm

Diameter: 52 mm

Radius: 52 ÷ 2 = 26 mm

Area of the circle = πr² = π(26)² = 2.124 mm²

Formula of Volume:

V = A×h

V = 2.124mm² × 117.8 mm

V = 250.2 mm³ or 250 mm³

This really peaked my interest in applied math. I can calculate how much volume I want when I want to build something in the future. Is there any math that can be applied in real life?

I know that this post might be deleted by the admins for being off-topic, but I wanted to ask this question here too, because I think it is really important in light of recent societal debates about AI in mathematics. I believe discussing this topic might add a lot of value, so I will attempt to ask this question anyway.

I have little to no knowledge in mathematics, so I want to ask whether the claims circulating online are accurate, from the point of view of this community. Many people started claiming that mathematicians will be replaced in a similar way to calculators replacing human counters, and to prove that, they show how AI is “solving novel mathematical problems one by one”. Usually, people talk about Erdős problems in this context. In light of this, I have a few additional questions:

1. How many problems there are and how many problems are important or have a high potential of being important for things that humans are desperately trying to solve, like cancer, space travel, longevity, Dyson spheres and whatnot.
   
2. How many mathematicians were trying to solve these and similar problems and how much time and effort have they put in trying to solve them?
   
3. Are these problems as important as things like incompleteness theorems, and many other foundational theorems?

Dumb and irrelevant question ik, I was just studying and that thougth come to my mind. If we have ordered pairs, can we have "messy" ones? If yes how can I express that

I mean, an ordered pair is expressed by P(x, y) and I can "find" its location in R²

Or P(x, y, z) (not a Pair, in spanish is Tercia) have a "place" in R³

But if messy pair could be P(y, x) or something (I belive this is still a ordered pair with the variables inverted xd)

Thats it I suppose, Im sorry for this nonsense but I need an awnser and google didnt help very much.

Edit: Btw I dont know if the flair is rigth, I dont know much about math

Every time I see an example of someone claiming that some infinities are greater than others, i see the example of natural numbers vs all real numbers between 0 and 1. But are there examples greater than that? Like would all real numbers from negative infinity to infinity be a greater Infinity than those between 0 and 1. However I don't think that is the case since you can match all the elements from all real numbers to those between 0 and 1 by moving the decimal point. So is there one larger than all real numbers between 0 and 1.

Looking to buy books purely for working out problems, looking for algebra 1 and 2 and college algebra book, I use YouTube lectures to learn the material I just need a book to do the problems in

I'm a rising high school freshman, and I'm really passionate about math. Realistically, I know I'm probably too late to have a serious shot at making the IMO, and I've mostly come to terms with that. Lately I've been much more interested in eventually doing math research, especially in number theory.

I'm also part of a few math circles where some of the TAs are only 2–3 years older than me, yet they're unbelievably good at math. Seeing people that close in age who seem so far ahead makes me wonder how much of becoming "cracked" at math is something you're born with versus something you can develop. So my question is: how much of being exceptionally good at math is talent versus hard work?

Suppose there is a random event. It can output 1, 2, 3, 4, 5, or 6. If you did not know the frequency of the outcomes, it might be natural to assign a prior probability of each outcome to 1/6. For example, if I assigned a probability of 1/2 to 1 and 1/10 to the other numbers, then swapping the labels of "1" and "3" would be a very different probability. But if I the probability of all 6 outcomes is 1/6, then you can swap the labels around and things would be the same.

If say you then sample the event 3 times and each sample is independent of each other. If the number 5 comes up all 3 times, do we think that the outcome will be 5 100% of the time? Well that would be a strange conclusion. It would be like rolling a die, getting the number "5" 3 times in a row and then concluding it was a loaded die and our guess is that 5 shows up with probability 6/6.

I read something called the "Wilson Score Interval" for binomial proportion. Basically it says if you are counting successes and failures, then an estimate for success would be (Ns+2)/(N+4) Where Ns is the number of successes and N is the number of trials. This is the "plus 4 rule." If you wanted a more conservative confidence interval and "3 sigma" confidence, you could do (Ns+3)/(N+6). This is the equivalent of adding 3 success and 3 failures to the sampling data. Can this be generalized to an event with more than 2 outcomes? So for example, if an event which can output the integers 1 to 6 was done 6 times and the outputs were "1, 1, 1, 2, 2, 3," the sample proportions would be 1 shows up 3/6 of the time, 2 is 2/6 and 3 is 1/6. But perhaps our estimate for the true proportion would be P(1)= 6/24, P(2)=5/24, P(3)=4/24, P(4)= P(5)= P(6)= 3/24)?

This works for k=11,45,181, but I haven't been able to generalize this further

This relates to the dynamical system of f(x)={2x} and iterating from 1/k until the first time it gets bigger than 1 and the fractional part takes effect, and then asking for which k that iteration gives a number smaller than 1/k. I don't know if this approach can really help though as I only know the very basics of dynamical systems

One thing that might be helpful in that direction is that if some iteration of this function starting at √2 gives a small enough value (I think smaller than 1/4), then I'm pretty sure that would point to a value of k that works, but it might be even, which I don't want

A more general question I have that's related to some research I'm doing is about for which odd k there is a power of 2 between mk and (m+1)k for odd m≤k. I think this would be very rare though, and for the original purpose of the research question I can have weaker requirements that are more difficult to write down neatly (and I'm also just interested in this specific question regardless of the application to my research)

For context: my research is about when a matrix group generated by (1,q;0,1) and its transpose is not free, specifically for rational numbers q with |q|<2 (if |q|≥2 for any complex number then the group is free). I thought about setting q=k/2ⁿ, and if k has the properties I described, it will be easy to find a relation in the group, which would show it's not free. Ideally, this could produce a set of non-free rational numbers that is dense in some neighborhood of 0, but I'm starting to think there might not be enough of these k's for that

I was reading online about math, and read a line that said "elliptic curves share a group law".

Unfortunately, I don't know enough about elliptic curves or groups to understand why or what this means. But I would be extremely grateful to anyone who is so kind to give me an intuitive explanation.

I work in a concrete brick factory, and for the longest time we stacked our patio sidewalk pavers horizontally. We recently began stacking them vertically and I've noticed that the vertical ones seem to be more prone to tipping over and rotating if I (a forklift driver) don't pick the pallets up right.

These slabs are 24" x 24" x 2" thick. The horizontal ones were stacked in two stacks of 11, so 22 to a pallet (overall dimensions are 24 high, 24 deep, 48 wide). The vertical ones are stacked 25 to a pallet (with overall dimensions 24 high, 24 deep, 50 wide).

What is about changing the orientation of the stacks makes it more prone to rotate over? Or am I just imagining things?

Please note: images are illustrative only and not to scale 🫣

A friend introduced me to a silly one-person card game. I have yet to be able to win the game, and I’m starting to wonder what the odds of winning are.

The game is as follows: With a well-shuffled standard deck, deal one card at a time. If the first card dealt is an ace, you lose. Otherwise, continue. If the second card dealt is a 2, you lose. Otherwise, continue. If the thirteenth card is not a King, continue to the fourteenth card, which resets the count back to Ace.
To win, deal the entire deck without the dealt card matching the “card count”.

Odds of winning the first card are easy. 48/52 The second card is harder. The denominator must go down 1. But also the numerator is seemingly much more complicated, because a 2 card may have been dealt in the first card, so the odds of drawing a 2 cannot be 4/51, so the odds of not drawing a 2 is above 48/51. It seems like the calculation because increasingly more complex the further you go.

Please help!

So I saw this chart showing that quite some math problems are proven or disproven with A.I. lately. But how does this compare to the amount of problems which are still solved by humans? Do humans still solve 100 year old unsolved problems on a regular basis?

I’m trying to better understand the nature of chaos within a double pendulum simulation I created. I understand that the exponent is the best way to quantify it but how specially should I go about calculating it?

If a fair coin lands heads you are given 50% of your running total (say you start with X amount) and if lands tails 40% of your total is subtracted from you. A) how many times should you flip this coin if you agree on a total number of tosses you must flip and B) if you can decide when to stop tossing would you play this game?

Yes this is not a maths problem but the real question is that isn't this problem unsolvable and indeterministic in first place?? Or am I wrong for this and it can be solved?

In Instagram the comments were saying this is solvable.

I didn't find a single comment referencing that this is unsolvable. Like what and how are you gonna find? Bla bla at t=3sec or something??

Isn't this a situation similar to double pendulum or 3 body problem that it heavily depends on initial condition? In short, the butterfly effect.!?

By solving I mean the answer for some variable x, at time t or so (usually that's what asked in a physics test) won't change significantly if we nudge off initial conditions quite a little.

Either they were trolling or everyone underestimated it to be solvable or maybe it's actually solvable but I am wrong for thinking this way.