Induction (extra) - Numberphile 

😂

If you can solve the first share with the world bro. Pretty sure that's the unsolved one!?

@@ephemera2 he’s just making a joke chill man 😭😭😭

@@dabluse3497 I know he's trying to but it makes no sense and the biggest problem in this world is how rare logic is and in turn people listening to those who sound smart but speak nonsense and lies i e politicians. His "joke" is like one of those "what's the difference between" jokes but he only gives a single object for comparing and no punchline because he's trying to tell a joke that he didn't understand but everyone else around him laughed when he first heard it so he tried to memorize it and doesn't even understand that you need two objects to compare the difference. And as you can see someone, aman, wants to understand something he doesn't completely understand at the moment and even try to give it a shot with an answer. People like this are toxic just like Faith Healers psychics and mediums. And now we have a world where half the population believes in astrology and horoscopes and determine what they should and shouldn't think about logically based on social taboos and social norms. In fact, since the entire world is run based on the decisions made and the words spoken by those who try to sound smart whether they are or aren't, so that people listen to garbage instead of thinking, the sway of ignorance has spread so far out of control that our scientific community is affected by it demonizing the truth and propagating what makes money.

@@ephemera2 Dude, you might need medication or therapy. This is the comment section of RU-vid, not a math journal. Lighten up, have some composure, and let other people participate. You're embarrassing yourself in trying to put down someone for misrepresenting the Riemann hypothesis. Do you realize that?

@@nathangreene3 I believe the first one is false, but unfortunately the YT comment section is too small to fit the counterexample.

Just flip the sings for getting all negative integers :)

direct proof for 3: odd integers: for n=1 we can take n = 1^2 for n≥1 we can take 2n+1 = (n+1)^2-n^2 even integers: for n=2 we can take n = 6^2-5^2-3^2 for n=4 we can take n = 2^2 for n≥3 we can take 2n = n^2-(n-1)^2+1^2 this also shows you only ever need at most three different squares with 2 = 5^2+11^2-12^2 we see that we also only ever need one negative

For question 2 I just tried replacing n with 41, which then obviously becomes divisible by 41: 41^2+41+41 = 41*(41+1+1) = 41*43. Didn't know that it always breaks for d-1 as well though, thats interresting!

@@SmileyMPV You are supposed to sum/subtract squares starting from 1. So you cannot say generate 4 by taking 2^2, because the sum doesn't contain 1^2. I had this problem at a national math olympiad competition many, many years ago. I could not solve it, but I was definitely not trained for it.

@@pieffe8 I guess that does make the problem more interesting. But as far as I can tell, this was not explained in the video.

For n² + an + b to produce a prime, starting with n=0, b must be a prime, that's all I can tell. n² -79n + 1601 produces 80 primes for the consecutive values 0 ≤ n ≤ 79.

what's the mathematical reason for the polynomial to give primes? (just fyi, it doesn't work for n=40)

@@Czeckie Ah thank you! Apologies, I should have said prime for 0

@@gammaknife167 thanks! this sounds interesting. I won't force you to give proof in yt comments. It seems to me that with this setup it should be an ideal norm argument but I'm way too rusty to work it out.

The first prob is Goldbach's conjecture, one of the most famous open question in number theory because of its involvement with Euler

the proof was that you could get every number from using ALL squares from 0 to n

1 + 1 = 2

What? The sum of any two odd numbers will be even, yet no even numbers (except 2 are prime), therefore no primes are the sum of two odd numbers (except 2).

@@MichaelWarman Precisely - _every_ even prime is the sum of two odd numbers. The fact that there is only one even prime makes it easy to check. This crops up in Gödel, Escher, Bach by Douglas R. Hofstadter.

@@AnOldGreyDog Yep. That's where I stole it from!

@@PhilBagels a brilliant book. Something worth stealing on nearly every page.

The first counter example is 40. But for 1 to 39 it works like a charm.

Yeah, and also -1+16

The provable one is amenable to an Inductive proof. [Note that Zvedelina warned not to spend too long on the first problem. That's the one that's a conjecture.]

Represent four as *4 = −1² −2² + 3²* and you can say that: Any *n* ∈ ℕ can be written as a signed sum of _consecutive_ distinct squares from *1²* up. It's much nicer imo.