Alternatively, one can define a prime to be any integer p such that Z/pZ is a field.

(However, fields are (by definition) rings with 1\neq 0. So implicit in the definition is p\neq 1. On the other hand, 0 and 1 are implicit in the construction of the positive integers too. So I like my answer.)

[QUOTE=Zeta-Flux;99654]Alternatively, one can define a prime to be any integer p such that Z/pZ is a field.[/QUOTE]

Is there an echo in here? ;)

Is there an echo in here?

Wow! ewmayer, take it as a complement that I missed you post, because your idea is the best one. :)

[quote=Zeta-Flux;99662]Is there an echo in here?

Wow! ewmayer, take it as a complement that I missed you post, because your idea is the best one. :)[/quote]

You mean compliment

No, compl[i]e[/i]ment.

One missed the post; the other posted the missed.

From Merriam-Webster (online):
[quote=http://www.m-w.com/cgi-bin/dictionary?va=complement] Main Entry: [B]com·ple·ment[/B]
Pronunciation: 'käm-pl&-m&nt
Function: [I]noun[/I]
Etymology: Middle English, from Latin [I]complementum,[/I] from [I]complEre[/I] to fill up, complete, from [I]com-[/I] + [I]plEre[/I] to fill -- more at [SIZE=-1]FULL[/SIZE]
[B]1 a[/B] [B]:[/B] something that fills up, completes , or makes perfect [B]b[/B] [B]:[/B] the quantity, number, or assortment required to make a thing complete <the usual [I]complement[/I] of eyes and ears -- Francis Parkman>; [I]especially[/I] [B]:[/B] the whole force or personnel of a ship [B]c[/B] [B]:[/B] one of two mutually completing parts [B]: [SIZE=-1]COUNTERPART[/SIZE][/B][/quote][quote=http://www.m-w.com/cgi-bin/dictionary?va=compliment] Main Entry: [B]com·pli·ment[/B]
Pronunciation: 'käm-pl&-m&nt
Function: [I]noun[/I]
Etymology: Middle French, from Italian [I]complimento,[/I] from Spanish [I]cumplimiento,[/I] from [I]cumplir[/I] to be courteous -- more at [SIZE=-1]COMPLY[/SIZE]
[B]1 a[/B] [B]:[/B] an expression of esteem, respect, affection, or admiration; [I]especially[/I] [B]:[/B] an admiring remark [B]b[/B] [B]:[/B] formal and respectful recognition [B]: [SIZE=-1]HONOR[/SIZE][/B][/quote]I think that compliment is correct, because it being complement in the way Zeta-Flux said it wouldn't make much sense, but it being compliment would.

[quote=cheesehead;99672]No, compl[I]e[/I]ment.

One missed the post; the other posted the missed.[/quote]

And who's pissed the most?

[QUOTE=davieddy;99676]And who's pissed the most?[/QUOTE]

:lol: Alex K. is probably kicking himself for letting you beat him to posting that zinger. (He PMed me the same quip.)

In any event, perhaps the fact that your last name is "Eddy" tells us the answer to your question. "Let it flow through you you must, young Whiz-taker..."

If I ever need a dose of humble pie, I know where to go!

(Now you can talk about mixing metaphors.)

[quote=Mini-Geek;99673]I think that compliment is correct, because it being complement in the way Zeta-Flux said it wouldn't make much sense[/quote]I was trying to show [I]why[/I] Zeta-Flux's "compl[I]e[/I]ment" did make sense ([b]1c[/b]) :geek: , but my alliteration and brevity seem to have obscured my meaning. :surrender

[QUOTE=S485122;99602]Mally,

I like to believe that all participants to the thread so far are well aware of the definition of a prime number. But you overlooked the purpose of this thread. The original question was :[/QUOTE]

[QUOTE=Mally] 2) A positive integer is prime if only *two* but only *two* distinct factors, are itself and unity

Euclid books 7 and 8 regards a number as a line interval compounded of units and
defines a prime as a number which can only be measured by the unit (not itself a number)
It follows from both the above two definitions that unity is not a prime[/QUOTE]

Well S4852122:

I admit I faulted in my second def:that's why I emphasised the word Two

In the next line I clarified grandscorpion's post by giving it straight from Euclid

Do all posters know this?

In the final line I gave a very reasonable explanation of why 1 is not considered a prime and this conclusion cannot be reached with out the use of the word two in the first def:

If you re-read the original problem the restriction is not only on 0 ,1 ,or 2 but *any other specified integer* Well where does that lead us?

Moreover my aim, as always, is not to appear a wisecrack with witty answers but to disseminate knowledge, fundamental at that, so people can pick up the bits and pieces of gems thrown in. Unfortunately the modern generation lacks a firm base and foundation!

Mally.