## Factors of Prime Numbers

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Message 1 of 2 , Dec 5, 2008
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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>
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• ... When we speak of prime numbers, without any other context, the natural numbers are implied so the above is indeed the case. ... Of course when you switch
Message 2 of 2 , Dec 5, 2008
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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.

When we speak of prime numbers, without any other context, the natural
numbers are implied so the above is indeed the case.

> However 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. Apparently
> this was all proved by Gauss.

Of course when you switch number systems, the definition of prime number
changes (and often becomes meaningless). It does retain its meaning
though in the Gaussian Integers (a+bi). Just Google that phrase
"Gaussian Integers".