@KannappanSampath yes

@KannappanSampath the last form is probably easier to work with for this problem

@robjohn When people say "only if" in everyday English, don't we assume they mean that if this condition and only this condition is satisfied, then the conclusion will be true? so in effect they are stating a bi-conditional, right??

@Former_Math_Addict not really. there are some unspoken contextual cues that imply a biconditional, but they are not strictly accurate.

@robjohn Are you already seeing a solution, Rob? I am stuck. :(

@Former_Math_Addict people often set legalistic traps that way

@KannappanSampath if $n = p$, then $\varphi(n)=p-1$

@robjohn Yes. Agreed. That is how, I said, $n$ is not a prime.

@robjohn Could you give me an example please?

$n-\varphi(n)\lt n\sum_k \frac1{p_k}$

afk

@robjohn Is this claim a variant of \eqref{1} or true in general?

oh, never mind. it is not true for $8$ and $p^n$ in general for a prime $p$.

@KannappanSampath It is true if there are more than one prime

There is equality if there is one prime

@robjohn I don't see a proof of this as well. :(

@KannappanSampath I will have to think for a bit

I will go back to work for now, and post a solution if I arrive at one. Please do post yours when you get to one. Thank you for taking time for me. @rob

@JasperLoy Hey Jasper :-)

@JasperLoy Hi!

@JasperLoy Yes, I'm looking for examples...

@KannappanSampath I haven't gotten anything yet, but I will keep looking

@JasperLoy The classic "if and only if" example given in Algebra is the Zero-Product Property. What I'm looking for is examples in everyday English for comparison.

...and contrast.

@PeterTamaroff Look at @BrianM.Scott 's solution you did not want to believe mine!!

Hi @BenjaLim

@KannappanSampath hey

I saw your number theory problem above

note that $n$ cannot be prime for this contradicts $\varphi(n) < n-1$

Yeah. noted.

But no further progress dude. :(

so we can assume it has more than one prime factor

@KannappanSampath funny thing is why is there $n^{2/3}$

True dat. That is mysterious.

@KannappanSampath Random fact about $\varphi(n)$: It is greater than 2 for $n > 6$.

@KannappanSampath I think

If we can show that having 3 or more prime factors leads to a contradiction

then it can only have 2

@BenjaLim I am trying to prove the linked assertion.

and then somehow show the exponents of both of them have to be 1

How does one make this work?

hmm

@KannappanSampath functions.wolfram.com/NumberTheoryFunctions/EulerPhi/29

@BenjaLim Heh. :)

what?

Yeah, it is terrible. How am I to know all these inequalities?!

I understand :(

@BrianM.Scott I told peter that he was over complicating the problem :D

@BenjaLim I just dropped in to ask: which problem was this?

the one on open sets in the product space

@BenjaLim I thought that that was probably it.

what was?

To be fair, it’s not uncommon to overcomplicate things when you’re just starting.

true

Yippie! I finally got the "Outspoken" badge!

well done!

cool

So people from three rooms will hunt for you!

Will Hunting!

When you change names without notice!

yoyoyoyoyoyoyo

@JasperLoy can you vote to close?

Close votes please.

"What do you mean by that? I'm not a native English speaker." - hmm...

@BenjaLim closed

@robjohn wassup dawg

@BenjaLim just fixing up an old answer that I should have fixed up a long time ago.

@robjohn do you surf?

@BenjaLim How about you?

@robjohn just about to head off to bed

ya know what

I just bought a pressure cooker!

@BenjaLim nope, nor do I ski or snowboard :-)

@BenjaLim those can be useful

@robjohn and a hand blender

google.com.au/imgres?imgurl=http://…

@BenjaLim Do you not have a regular blender?

@robjohn a pressure cooker is really handy

@BenjaLim or is this more convenient?

@robjohn no. This one is easy, you can blend stuff in the pot you cooked them in!

@BenjaLim Ah, neat.

@robjohn do you cook much?

I absolutely love cooking

@BenjaLim I used to more than I do now, but I do once in a while.

awwwwwww

you should man

Like the other day I was in Geelong

I had a gnocchi, peas and lamb thingy

@BenjaLim Then I would eat more, and I can do without that :-)

Tasted it, made a few guesses of what some of the rest of the ingredients were

and cooked it, so tasty!

@robjohn weight problems?

@BenjaLim nope, and I'd like to keep it that way.

no cholesterol?

would love to go to american one day...

@BenjaLim Just a bit, but it's under control with a minimal dose of statin

@rob have you been to australia?

@BenjaLim Nope, but it is one of the places I would like to go.

you're missing out!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

@BenjaLim I know a lot of geography there from some board games we play.

beautiful beaches, beautiful girls (or guys) by the beach, sun, blue sky......

@robjohn as in?

@BenjaLim Australian Rails

oh my I have never heard of this game :D

@BenjaLim It is one of a number of crayon-based rail games

gawd....

@robjohn capital of australia?

I confess my knowledge of US geography is not very good.....

@BenjaLim Canberra

largest city?

what are the blue mountains?

I had this big debate that day with some guy

on how to pronounce yosemite