<p>can prime numbers be negative?</p>
<p>By definition, prime numbers have to be greater than 1. </p>
<p>A prime number is divisible by only itself and 1, only two factors. </p>
<p>If you have a negative “prime” number such as -5, then the factors would be 1, 5, and -1 and -5. (4 factors)</p>
<p>No, wouldn’t the factor of -5 be only -1 and -5 because 1 and 5 are the factors of POSITIVE 5? Therefore showing -1 only has two factors?</p>
<p>-5 * 1 = -5
-1 * 5 = -5 </p>
<p>All four numbers would be factors. </p>
<p>When I say that 1 and 5 are factors of -5, it does not necessarily mean that 1 and 5 have to be the ones multiplied together. You are able to multiple 1 and 5 by something to get -5. (as I showed above)</p>
<p>just trolling but technically, using your logic 5 wouldn’t be a prime number? factors can still be -1, -5, 1, 5.
1<em>5 = 5
-1</em>-5= 5</p>
<p>Haha, got me there. I think I tried to over-explain.</p>
<p>Prime numbers can ONLY have factors of one and itself. </p>
<p>There! :)</p>
<p>You had it right when you said that it was part of the definition, but it’s a part that often goes unsaid and then gets forgotten. Prime numbers are positive integers whose only positive factors are one and themselves. The possibility of either negative primes or negative factors being used to exclude numbers from being prime is ruled out by definition. You don’t have to know the reason why just like you don’t have to know the reason why 1 is not prime. (The briefest explanation is that if the definition didn’t work that way, then many theorems begin: “If for all prime numbers greater than 1…” rather than just “For all prime numbers…”)</p>
<p>(Disclaimer: The material below probably won’t be helpful for the SAT.)</p>
<p>Just a remark: In general, a “prime” in some ring is an element p such that
(1) p is nonzero,
(2) p is not a unit (i.e. there is no element p’ such that pp’ = 1),
(3) whenever p divides ab, either p divides a or p divides b.</p>
<p>If the ring is Z, the integers, then the primes include …, -7, -5, -3, -2, 2, 3, 5, 7,…</p>
<p>You’re right in that it’s common not to distinguish between primes up to “multiplication by a unit” (a unit is an element u such there exists some u’ with uu’ = u’u = 1 – i.e. 1 and -1 in the case of Z, the integers). So one often treats 3 and -3 as “the same” prime. This is because they generate the same prime ideal…</p>
<p>This ended up sounding borderline pedantic, and that was not the intent. Just thought that someone might be interested in the formal resolution of this question.</p>