Product of two squares is a square.

It is relatively easy to show that product of a square and a non-square is a non-square.

We know the number of squares and non-squares, which leads us to the conclusion that the product of non-squares is a non-square.

(I should probably have written everywhere quadratic residue and quadratic non-residue, but square/non-square is shorter.)

(I wonder whether this is mentioned in some post on the main.)

@MartinSleziak, it's the third case you mentioned that I'm having trouble with. And I think you wrote it wrong: the product of two non squares is a square. Since $\displaystyle \bigg( \frac{a}{p} \bigg) \bigg( \frac{b}{p} \bigg) = (-1)(-1) = 1 = \bigg( \frac{ab}{p} \bigg)$?

My professor mentioned something about looking at the prime factorization and then the powers should end up being even (or something along these lines; it made sense when we were discussing it, but when I tried to replicate it on my own, I couldn't)

@RobertCardona r u there

Can I ask u something

My argument is the following. We are working modulo $p$, $p$ being an odd prime. We know that among the numbers $\{1,\dots,p-1\}$ there are $(p-1)/2$ squares and $(p-1)/2$ non-squares.

Imagine multiplicative table of $\mathbb Z_p^*$.

Each row will contain all values, in particular the same number of squares and non squares.

So in the table with $(p-1)^2$ elements, half of them are squares and half of them are non-squares.

What we have in this table?

Some entries are product of two squares. They are squares and there is $(p-1)^2/4$ of them.

Then we have entries which are obtained as a product of non-square and a square. There is $(p-1)^2/2$ of them and all of them are non-squares.

So since there are already $(p-1)^2/2$ non-squares in the table, all the remaining fields must contain squares.

So the remaining entries, which are obtained as product of non-squares, contain squares.

Does this counting argument make sense?

@RobertCardona Of course, you are right. That is what I had in mind. Product of two quadratic non-residues is a quadratic residue.

@RobertCardona It is interesting to hear that there is also a different argument. I wonder whether something about this can be found on the main (or on some other sites).

Some proof of the fact that product of a quadratic non-residues is a quadratic-residue is also given in this answer:

It seems to rely on the fact that there is the same number of residues and non-residues, too.

@SayanChattopadhyay, hi!

@MartinSleziak, thanks for the link, I came across that one before I asked in chat.

I only looked briefly at the solution, but the argument seems similar to mine.

I was looking for the same argument I had seen myself

thanks.

what you said makes sense to me, thanks!

Well, this is the only one I am aware of (apart from Euler's criterion).

I probably need to digest it a little to make sure I can think it through on my own next time I come across something like this.

Maybe somebody will notice the discussion here and will make another suggestion.

that would be good!

Or maybe you will have a possibility to talk to your professor again.

I could just ping him on here, but I'll try to talk to him in person first.

Oh, he's on MSE. That's nice.

hi guys, wow, an actual conversation in here :)

fyi you cant chat ping users unless they posted in the last 7 days. there is a way to issue an invite though. (rather hidden/ obscure)

Anyway, I've already said about the problem what I know. We will see whether somebody else will be able to contribute something reasonable.

How vzn?

Thanks again Martin!

it would be interesting to see that though. few from MathOverflow stop around Mathematics....

ok its not hard, you have to be entered into the chat room for the invitation (which you are, here). then you look up their chat user & click on it, & the button "invite to [x] room" should be present.

@vzn You mean this? Inviting a user to chat from the profile page?

@RobertCardona my question is

Hi!

MS yep thats it. not documented in the chat FAQ afaik. some se doc is spread in meta at times.

(You can always ping a user on some of their posts and remove the comment later. That seems simpler to me, although it should be used in moderation.)

Can the cube of every perfect number be expressed as the sum of three positive cubes

Pls someone help me with this proposition

@RobertCardona u know how to do it

not off the top of my head.

isnt that a special case of eulers conjecture?

What?

which was proven false for n^4.

No idea @vzn

not sure the case for cubes.

sayan why are you interested? did you post this as a question?

No a perfect number

Yes I did

The question is here.

I didn't get the proof

lol (sigh) there are cases listed.

So any ideas

it seems to show a rough statistical amt of cases succeeding & gives individual cases.

So vzn is this a difficult question to answer

I found this question while doing my maths exam

what exam were you working on?

So is this right@vzn

is what right?

My daily maths exam

My proposition

its a difficult question but see nothing wrong with the answer from TP.

So can u answer

But that's not the proof

Well I just need some points

I will try to get the proof myself

did this exact problem show up on your exam? afaik it is probably an open question in number theory.

are you in a number theory class?

No.....I am a 10 grader.....I was doing a question on surface areas and volumes.

ok, looking closer, TPs answer is not a complete answer to the question.

but shows good analysis.

So we require something more complicated

the question is not very precisely stated. people are asking about the domain. negative numbers/ rationals etc... assume those are ruled out because they are not perfect...?

Well @vzn is this related to the odd perfect number conjecture

Positive numbers

how so related to odd perfect number conjecture?

Coz if there r no odd perfect then only even perfect are present

as someone asked the 1st task is to show it holds for low perfect numbers via computer search.

otherwise what gauss said re FLT comes to mind.

It holds good TP did try it with his computer

if TP did a computer search to show it hold for low perfect numbers, he did not explain it very well in his answer, dont follow/ see that.

Yes .......but then if it is holding good for most of them it should for all of them

What is this FLT

U got something @vzn

excerpting quote from sophie germain / simon singh

> Gauss’s work influenced every area of mathematics but strangely enough he never published anything on Fermat’s Last Theorem.

> In one letter he even displayed contempt for the problem. His friend the German astronomer Heinrich Olbers had written to Gauss encouraging him to compete for a prize which had been offered by the Paris Academy for a solution to Fermat’s challenge:

> “It seems to me, dear Gauss, that you should get busy about this.” Two weeks later Gauss replied, “I am very much obliged for your news concerning the Paris prize. But I confess that Fermat’s Last Theorem as an isolated proposition has very little interest for me, for I could easily lay down a multitude of such propositions, which one could neither prove nor disprove.”

He envied Fermat that's what I think

history very much bears this out. despite celebrated recent progress re twin primes conjecture, there is not even a solution after 2 Millenia!

same for the odd perfect # conjecture!

gauss had no reason to envy anyone :p

my suggestion, learn about eulers conjecture & look up the code that was used to solve the 4th power case, & try fitting it into your question.

start by compiling a table for low n. have not even seen this so far in work by you or TP although it seems to touch on the idea.

U know codes

(lol) yes.

This is why number theory is so amazing

U can try

yes! (re number theory)

Yes?

no have too many other conjectures to work on.

Oh got it....well this is a legit question right@vzn

absolutely it is "egitimate" even bordering on interesting if its true for all low n searched but many basic number theory questions are unanswerable by experts.

what have you studied in number theory so far?

I have done till modulo

Congruency

ok good, thats something

U r an expert @vzn

there are a bunch of refs to computational number theory in this chat room, check it out.

Who are they

do not regard myself as an expert only a motivated amateur/ enthusiast.

what book did you study for number theory?

Elementary number theory by David m burton

elementary number theory / burton

how much of it did you get thru?

But my childhood dream is to prove the Riemann hypothesis

Till modulos

(too bad no look inside/ TOC on amazon for that book)

ah there is a lot about riemann in this chat room so far.

only a mere ~1½ century old, a mere youngster in comparison. at least that one has a $1M incentive on it. :)

So how should I get the codes for my question

Well I m doing it for obsession not the money

obsession is likely necessary but not sufficient :p

do some research on eulers conjecture.

I will for sure......

its a great story (in number theory). it was not disproven until computer searches found the counterexample for 4th power case.

the code to search for your variant would likely be very similar.

but even somewhat "naive" code would probably succeed on checking low n.

Which computing language should I use

scripting languages are very good for rapid prototyping. either python or ruby. python has good libraries now. have used ruby myself many years.

eg check out my blog re collatz conjecture & always looking for volunteers wink

@vzn can u help me I will be very grateful to u

java is good for bignums (ruby does fairly well on that too). java also good for faster performance & eg parallelizing (multithreading) etc

←helping you now

No codes pls....

are you in high school?

college?

So that I can take this question to the highest institution in india

High school

The professor told me to get out of his office coz my question is very easy for him

fyi BS who has chatted quite a bit in this room (not recently) knows a lot about riemann. some few refs to his writing here

the professor is presumably mistaken. & is apparently not very friendly.

Yes.....

Well low perfects mean till which perfect number?

your question does touch on existence of odd perfect #s. (re 2M old)

Hos

How?

? you claimed that yourself. simply agreeing.

posting this link for the room. plz star it!

I m back

elementary number theory, burton / Barnes & Noble incl TOC

wow maybe 1st published ~1976 acc to google

ah look at that FLT sec 12.2

nice fairly broad coverage!

eg looks like it covers RSA, sec 10 cryptography