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I was always led to believe that the factors of any prime number are itself=

and one, and of course the relevant negatives, and no others.=0A=0AHowever=

the other day I came across a note that claimed the prime number 5, for ex=

ample, has the factors (1 - i) x (1 + i). Presumably other prime numbers (a=

ll?)=A0have higher values of the square root of -1 in the equation. Apparen=

tly this was all proved by Gauss.=0A=0ACan anyone guide me to any more know=

ledge on this topic to find out how this factorization is arrived at?=0A=0A=

Regards=0A=0ABob

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<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>I was always led to believe that the factors of any prime number are itself and one, and of course the relevant negatives, and no others.</DIV>

<DIV> </DIV>

<DIV>However the other day I came across a note that claimed the prime number 5, for example, has the factors (1 - i) x (1 + i). Presumably other prime numbers (all?) have higher values of the square root of -1 in the equation. Apparently this was all proved by Gauss.</DIV>

<DIV> </DIV>

<DIV>Can anyone guide me to any more knowledge on this topic to find out how this factorization is arrived at?</DIV>

<DIV> </DIV>

<DIV>Regards</DIV>

<DIV> </DIV>

<DIV>Bob<BR></DIV>

<DIV style="FONT-SIZE: 12pt; FONT-FAMILY: times new roman, new york, times, serif"><BR> </DIV></div></body></html>

--0-1715832665-1228474496=:81077-- > I was always led to believe that the factors of

When we speak of prime numbers, without any other context, the natural

> any prime number are itself= and one, and of

> course the relevant negatives, and no others.

numbers are implied so the above is indeed the case.

> However the other day I came across a note that

Of course when you switch number systems, the definition of prime number

> claimed the prime number 5, for ex= ample, has

> the factors (1 - i) x (1 + i). Presumably other

> prime numbers (a= ll?)=A0have higher values of

> the square root of -1 in the equation. Apparently

> this was all proved by Gauss.

changes (and often becomes meaningless). It does retain its meaning

though in the Gaussian Integers (a+bi). Just Google that phrase

"Gaussian Integers".

Instead of adding i=sqrt(-1) to the numbering system one could add

sqrt(5), so 5 becomes a perfect square. Or add sqrt(6), then 5 =

(sqrt(6)+1)(sqrt(6)-1).

But there are many pitfalls. 1 gains infinitely many different divisors

in some number systems (1=i*(-i) in the Gaussian numbers). Prime

numbers must be replaced by prime ideal in most systems...

Google "algebraic number theory," or better, look it up at your local

library and get a book or dozen books on the subject.

CC