For fun

How many factors of 2^100002+1 can you produce?

;-)

More fun.

Provide an example of a solar system or reasonable configuration star system where a small mass (eg spaceship or asteroid or small planet) orbits literally around nothing (as in the star or stars of the system are not inside its orbit)! Is it stable?

If not can we make it stable by adding solar panels to the orbiting mass and ionic propulsion to correct its orbit from time to time with only relatively minimal energy (to its overall energy) from the stars of that system?

Lagrange points of a binary star?

Tomorrow maybe I'll try to find and prove the complete set, assuming I wasn't lucky enough to find them all already.

Yes you can call it that like L4,L5 , some extreme version of it, because it is similar in concept but its funny that the usual ones still have the star inside them and that would appear to orbit nothing. Now on to details of orbit and stability hehe.

It has also 2 very big ones (almost same number of digits) that almost everyone will miss. Its remarkable situation that hard to imagine without the more general problem study of some special sum of squares.

By the way, IDK why I listed trivially composite factors in my first reply.

I'm not very strong in number theory, unfortunately. I doubt I'd be able to solve this, but I do have a question:

You may want to also think identities like x^(2n+1)+y^(2n+1)= (x+y)*(x^2n-x^(2n-1)*y+...+y^2n) and that 100002=2*3*7*2381 so it could be seen as sums of odd powers.

But then it gets interesting quickly and there are two factors that are same in number of digits too.

Now for only prime factors that is interesting but i havent worked it out yet.

That is a number of thousands of digits. So there is room for a lot of primes in there.

someone make me feel like an idiot, please. What will take me an hour to figure out, you guys will figure out in 3 minutes or less after setting up that OOO so beautiful equation.

I am deciding whether to auction my points for higher value or straight up buy a gift card. Now, if I am reading this stuff correctly the details are as follows: I can buy a $10 gift card for 1,000 points. I also have the chance to enter an auction for the same $10 gc. I can buy 15 entries for 125 points. It scales as follows: 10 tickets for 85, 5 for 45, 3 for 28, and 1 for 10. I am only concerned with the 15 ticket purchase for simplicity and because it is maximizing EV. The auction is capped at 135 participants(not sure if I am capped at 15 entries(will worry about this later )). Is my EV higher in the auction or getting immediate $10 for 1k points?

If it's 135 entries max, for 1000 pts you could buy 8*15=120 entries. So your 1k pts but you either $10gc or 120/135 chance at $10gc. Take the cash

guess I was looking for a points/per number. thanks for the dumbed down and make me even look more ******ed than I thought post

does anyone have an equation? There are many auctions. I need to look at some ratios to max my EV

i mean... you can do the same thing for any auction. Find best entries/pt, and use that to do what i did above. Even if you can't physically buy all 135 entries, ask yourself how many points it would cost to do so. If it's less than what it costs just to buy the GC, then go the auction route. Otherwise buy the GC direct. If you have multiple auctions you can enter that are more valuable than direct GC, then you see which ones cost the fewest points to buy the whole auction's worth of entries and max those in order. ez game

The odds are good that every single auction is -EV relative to buying straight up unless you can predict auctions that are less competitive.

Fun challenge; The prime number theorem claims that Np(n) or the number of primes up to n goes like n/log(n)

Imagine civilization collapsed and you know nothing about these theorems and someone asked you to produce bounds on primes like some function of n that gives an upper bound on primes or the number of primes up to n. What can you come up with using basic simple math arguments?

One can also produce claims like there is at least one prime between eg p and 2p where p is the last prime. Can you prove any such or similar claims?

At a point just after cut-off in a steam engine cylinder, the steam pressure was 1100 Kpa and the dryness 0.85. After expansion, at a point just before release, the pressure was 280 Kpa and the dryness 0.80. Assuming the expansion obeys a law pV^n=c, determine the heat flow per kg across the cylinder walls, and state its direction.

Nobody? These are practice questions for an upcoming professional exam. The answer given was 0.2 kj/kg. I can't come to that answer for the life of me. I'm thinking the answer is wrong but I was hoping someone smarter than me could confirm it.

Econometrics question. Probably not difficult. Could someone point me in the right direction on this one?



My question relates to the highlighted part. How do you determine the variance of the error term if you know R squared?

Defn of R-squared = 1 - SSE/SST = 1 - Var(Errors)/Var(Total)

where in the notation from above SSE is proportional to V(e) and SST is proportional to V(r).

So we have R-squared = 0.8 = 1 - V(e)/V(r)

and V(r) = 0.0081 + V(e)

Algrebra yields:

V(e) = (.2)(.0081)/.8 as shown in the text above.

Thanks a lot. That ratio part eluded me for some reason, but it makes sense now

Linear Algebra

https://math.stackexchange.com/quest...ynomial-of-t-v

At the end of the second paragraph of the answer, it says "Using minimality of the minimal polynomial show that P(T) is neither zero nor invertible".

I understand everything else in the first 2 paragraphs of the answer, but I do not see how to prove that P(T) is not invertible. Proving this is equivalent to proving that there is a non-zero vector in the kernel of P(T).

(Also, I am pretty sure that the minimal polynomial is not always equal to the characteristic polynomial, but it is equal to (-1)^(\dim V) times the characteristic polynomial.)

I would appreciate any help.

Never mind, I found the fact I needed.

If anyone is interested, it is here: https://artofproblemsolving.com/comm...s_on_wikipedia

(Note that I would need to take P to be irreducible.)

Explain this to me plz...

5
∑ 375(-1/5)^(i-1)
i=1

I got 312.60 amiright?

I think "IQ" is just nonsense. See https://medium.com/incerto/iq-is-lar...e-f131c101ba39.

Yes.