A tentative question
 

There seems to be no non-residues higher than quadratic order;is this related to Fermat's last theorem?

If I'm understanding you right, you're saying there are no, say, cubic non-residues. Well 4 is a cubic non-residue modulo 7 (there is no [i]x[/i] where [i]x[/i]^3=4 (mod 7)).

[QUOTE=Happy5214;550172]If I'm understanding you right, you're saying there are no, say, cubic non-residues. Well 4 is a cubic non-residue modulo 7 (there is no [i]x[/i] where [i]x[/i]^3=4 (mod 7)).[/QUOTE]

Exercise for the O.P.

If p = 1 mod 6, how many cubic residues does it have? how many non-residues?
What about p = -1 mod 6? Did you bother at all to even try to answer your own question?

Do us all a favor. Don't come back until you take the time to actually do some reading
about this subject. And if you can't be bothered, then don't come back at all.

Your question could have been answered with 15 seconds of thought. But of course
you could not be bothered.

[QUOTE=R.D. Silverman;550175][QUOTE=Happy5214;550172]If I'm understanding you right, you're saying there are no, say, cubic non-residues. Well 4 is a cubic non-residue modulo 7 (there is no [i]x[/i] where [i]x[/i]^3=4 (mod 7)).[/QUOTE]

Exercise for the O.P.

If p = 1 mod 6, how many cubic residues does it have? how many non-residues?
What about p = -1 mod 6? Did you bother at all to even try to answer your own question?

Do us all a favor. Don't come back until you take the time to actually do some reading
about this subject. And if you can't be bothered, then don't come back at all.

Your question could have been answered with 15 seconds of thought. But of course
you could not be bothered.[/QUOTE]I think you quoted the wrong person.

[QUOTE=retina;550176]I think you quoted the wrong person.[/QUOTE]

Huh? I did not quote anyone. I simply made a follow-on post. My post was addressed
to the O.P.

[QUOTE=R.D. Silverman;550181]Huh? I did not quote anyone. I simply made a follow-on post. My post was addressed
to the O.P.[/QUOTE]You quoted Happy5214 in the same way I'm quoting you.

You clicked on the "Quote" button, not the "Post Reply" button.

Technical point in that your text clearly addressed the OP, not the person you quoted in the technical sense.

[QUOTE=xilman;550190]You quoted Happy5214 in the same way I'm quoting you.

You clicked on the "Quote" button, not the "Post Reply" button.

Technical point in that your text clearly addressed the OP, not the person you quoted in the technical sense.[/QUOTE]

Point taken. This has become an interesting debate over the word "quote".

What [i]I [/i] wrote was a follow-on. It was suggested by Happy5214 that the O.P. check whether
4 mod 7 was a cubic residue. I simply extended this to other primes that were:
firstly 1 Mod 6 (as 7 is) and then to all other primes. [And added some admonishment.]


From now on I will try to use the "post reply" button instead of "quote" to avoid confusion when I do not
intend to quote anything.

[QUOTE=R.D. Silverman;550228]This has become an interesting debate over the word "quote".[/QUOTE]I have copied (some of) your text and placed into a box above. That is a quote of what you said. You didn't write my post, I wrote it, so I quoted you.

[QUOTE=R.D. Silverman;550175]Do us all a favor. Don't come back until you take the time to actually do some reading
about this subject. And if you can't be bothered, then don't come back at all.

Your question could have been answered with 15 seconds of thought. But of course
you could not be bothered.[/QUOTE]

I am quoting R.D. Silverman, but addressing the moderators here:

Why is this sort of rude, insulting tone permitted on the forums? Certainly this vitriol and condescension discourages people from asking questions and thereby learning. Even if the answers may be trivially obvious to Mr. Silverman, they are not to others.

[QUOTE=mathwiz;550233]I am quoting R.D. Silverman, but addressing the moderators here:

Why is this sort of rude, insulting tone permitted on the forums? Certainly this vitriol and condescension discourages people from asking questions and thereby learning. Even if the answers may be trivially obvious to Mr. Silverman, they are not to others.[/QUOTE]

The O.P. has a very long history of crank and nonsensical posts combined with
a proven unwillingness to spend time learning anything about the subject. He also
fails to respond to questions posed to him.

Anyone who wants to ask questions needs to show that they have made at least a
minimal effort to answer the question for themselves. To do so otherwise is rude
in and of itself because it places a time requirement on others. Even a cursory
web search by the O.P. would have revealed the answer. Teachers should not
have to waste their time with students who are unwilling to do basic homework.

Eh... There is a button for that.
 

[QUOTE=mathwiz;550233]I am quoting R.D. Silverman, but addressing the moderators here:

Why is this sort of rude, insulting tone permitted on the forums?.[/QUOTE]

As a purely gedanken experiment, I might submit that the opposite is also true:

[QUOTE]The comments like that (a comment from someone who has nothing for or against the argument and only stopped by to criticise the tone of voice in which two members who know each other for 10+ years talk to each other) - that discourages qualified answers. people who know the answers stop answering because someone will "report them", so they just stop. And then forum fills up with very helpful and kind answers from folks who don't know the answer and don't even know how to learn the answer.[/QUOTE]