I've just started a proofs course. I was told we're allowed to use the Fundamental Theorem of Arithmetic, that each integer greater than 1 can be expressed as the product of primes in exactly one way.

I understand this fact, but I'm not sure how exactly to use it in proofs. This is what I came up with, but it seems more complicated than necessary. I'm wondering if there's a simpler way.

Prove: If x is an odd integer, then x^(4) is an odd integer.

Proof: Suppose x is an odd integer, and suppose its prime factorization is x = p_1 · p_2 · p_3 · ... · p_n. Since x is odd, p_1, p_2, p_3, ... , p_n are all not equal to 2. x^(4) = p_1^(4) · p_2^(4) · p_3^(4) · ... · p_n^(4). Since the prime factorization of x^(4) does not contain 2, x^(4) is odd. QED

I know you could prove the same result by defining an odd number to be of the form 2n+1 witn n ∈ ℕ, but I just used this as an example because I thought it was something I could probably prove with the Fundamental Theorem. I just want to make sure I am using it correctly.

Thank you :)

I know it is useful in solving diophantine equations in some way, but I don't know how. Fermat's Last Theorem is a good example. Apparently the fact that Z[zeta_(n)] is not a UFD had something to do with its difficulty, so how does it being Dedekind help the situation?

I've been working on a nifty program that takes subsets of the integers and determines whether there is unique factorization (by computing primes up to a number and factoring).

For example, take the integers greater than 3. 4 is prime because it can't factor. So are 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (none factor). 16 is 4*4, so it's composite. Continuing this will eventually fail at unique factorization because 36 = 6*6 = 4*9, or 110 = 5*22 = 10*11, and many other examples.

Let's get weirder though. Take integer subsets of the form m*n+b, where n >= 0 and everything is an integer. For example, take 2, 5, 8, 11.... 2 is prime, 5 is prime, 8 is composite, 11 is prime, 14 is prime, and so on. This does not preserve unique factorization because 140 = 2*2*35 = 2*5*14 (and this is the first example!). Others, like n+2, 2n+3, 4n+2, 4n+6, 5x+7 (at least up to 160000*) (see here), all have unique factorization.

Is there a way to determine, given an m and b, whether or not the subset m*n+b preserves unique factorization? What are other subsets worth exploring?

Would the fundamental theorem of arithmetic be a perfect hash function, if in fact it is impossible for it to produce collisions?

I mean this in the sense that is used in Godel numbering: where every character is assigned a number, and then you take the product of successive primes to the power of the assigned numbers to produce a unique number.

Thanks.

If a and b are two natural numbers s.t. (a,b)=10 and [a,b]=100 what could a and b be? I get that there's the GCD and LCM, but I'm not exactly sure how/where to start solving the problem.

Edit: Thanks for the help, understood much more now.

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Pelton mentions that the Clippers were initially hesitant to go small, because they feared that Ayton would take advantage offensively. But the Zubac injury forced their hand, and it worked perfectly!

The Suns were outscored by 22 points in 37 minutes (+8 with Ayton on the bench).

The gawd Ty Lue will surely take advantage of this for the rest of the series. It’s truly a fascinating development - teams can spend #1 picks or max contracts on players like Ayton/Gobert, yet getting a Batum on the minimum might be just as good!

This discussion happens at around the 14:30 mark of the latest Hoop Collective podcast: https://open.spotify.com/episode/4s2KTB9gbzEWSsXCs4pg5c?si=59NLmN44TUSVHQnThxn94g&dl_branch=1

Think about it. DBD gameplay has the highest potential and opportunities for players on both sides(Though more for Survivors) to rub victories in their face which leads to a higher amount of players being sore winners and losers. Its the fact that the win is not instant and gives players on both side limited(or infinite) time to rub it in to the other side that they have won and there is likely nothing they can do about it. How is this so?

Survivors:

-When they are the gate and know they can safely win, it is possible for them to taunt the Killer and show them that they have won and they can do nothing about it at that point. Power over the Killer.

-When they have a key and are at the hatch that they can open instantly(before the nerf) or jump into instantly if its already open. There is usually nothing the Killer can do by then, allowing Survivors to rub their victory in the face.

-When certain conditions are met that basically lets the Survivor get a small win that usually the Killer can do nothing about, such as teabagging with DS on, being stunned after a pallet, body blocking and more.

Killers:

-When the Survivor is on the hook and the Killer rubs it in their face by showing how powerless they are by smacking or nodding at them. This can be offset by DS & Deliverance or friends coming to save, but its still time to rub the victory in.

-When the Killer slugs everyone on the ground and lets them bleed out. He could end it, but he wants to show the Survivors he has won and there is nothing they can do about it if they used up their unbreakables.

-Nodding while carrying or after downing a Survivor to rub it in their face.

DBD is unique as the gameplay is designed in such a way that when one side gains a victory upon the other, they have ample time to rub their victory in the others face without facing repurcussions if they are knowledgable. Some perks can offset this like Blood Warden, DS, Unbreakable, NOED and more.

Other games have designs that can't afford players to rub victory in the face of others after a victory though. For example in Siege, trying to rub a victory in their face can cost you your team if they shoot you. And if there is a win, it is already instant and then the match is moved on to the next round or ends which gives little time for the other player to rub it in except for posting profanities over text.

Other games give more risk to taunting, such as in TF2 where you can use a taunt animation a player after kil

For something that is so wildly different than any phone I've ever used, I am surprised by how quickly I've gotten accustomed to the novelty of it. Don't get me wrong, it's still cool as fuck. But after just a day of using it, it already feels so natural.

Edit: spelling in title - form* factor

The first question is pretty straightforward. I'm talking about Belle, Colette, Amber, Sproot and the like. As for the second, I mean as to whether the MAIN ATTACK (red button) has an impact on the brawler's viability somehow. For Belle, it basically carries her, and is generally superior to her super. But for Gene and Stu, it pairs with their supers and charge rates to make them great picks.

And I have to mention Surge somewhere, this is a u/s_raut post. So let's say that Surge's attack is unique, but mostly carried by his gadget and super charge rate.

So, who d'you think has a unique enough attack that it makes them good?

Have they given any info on that or do we have any data yet?

I'm guessing that there are no two natural numbers that have the same sum of square roots of their prime factors. I can easily rule out some cases, but don't know how to go about proving it more generally. Any ideas?

Don't worry not the Lu Bu hater lmaoo, I'm the factual supremacy guy hence why I'm Dio. The Legend of Lu Bu spinoff has really done more for his character and shown him as this goliath amongst normal humans, like a true monster similar to Kaido which they share a lot of similarities. However Lu Bu unfortunately really wasn't given anything special like as to what the god fighters and other human fighters have. I think his Sky Eater would wreck a lot of people's shit but that's his final move and ig Red Hare since in case his lower half is destroyed he can rely on him and keep attacking, which others cannot. But that's really it, which might be just how do to how early on stuff was even tho Thor was still very unique in his fighting style with hammer boomerang throws and lighting generation.

Tbh I see Lu Bu as the fighter you choose when you go up against an opponent you don't know at all since he's strong overall in all stats so you have the best chance. But once finding out their skills then he's at a disadvantage. His melee skills could help especially since the Halberd ranged but like Adam's eyes, Sasaki's double volund and analyzing ability, Jack's IQ, and Raiden's muscle control super strength. Lu Bu is second physical strongest to Raiden so maybe his melee would help beat Raiden, but again not fully sure.

For now he maybe best melee fighter but you have Sasaki and I lean towards Sasaki more. Also for everyone saying Brunhilde said Lu Bu is strongest human, no he's not, she said strongest in 3 kingdoms. And Poseidon has been called God of Gods, Shiva strongest god of destruction and those titles have some truth but mostly to hype up their reputation. The strongest humans will be Leonidas and Kintoki, and considering they are melee too, they'll be better than Lu Bu by default and have a unique power or skill otherwise the authors will be accused of making Lu Bu parts 2 and 3 and getting generic. Hopefully in this spinoff we see more of Lu Bu fighting rather than always just his reputation scaring people cause I wanna see more of his techniques and skills, but that aside forget the strongest human thing. Again rn he's kinda base, granted a very strong one and he'd put up high difficulty fights against a lot of people, just not win them imo. And compared to the god fighters..... yeah nah god fighters just all in a dif league rn with unique abilities by default.