Look at -7. Its only integer divisors are \pm 1 and \pm 7. In a superficial reading of "a prime has only two positive divisors," it looks suspicious — but -7 clearly behaves like 7 everywhere it shows up: -7 \cdot n is divisible by 7, -7 has no "interesting" factors, and if you list the primes and forget the sign, |-7| = 7 is squarely on the list. So why do textbooks insist that primes are positive? Is this just a convention, or is there a real reason?
The short answer: in \mathbb{Z} (the ordinary integers you meet in school), primes are defined as positive by convention, and the convention pays off. But the deeper mathematical picture — where -7 is prime in a certain sense — is worth understanding, because it shows up the moment you stop doing school arithmetic and start doing algebraic number theory.
The school-level definition
Your Class 9 textbook says something like:
A natural number p > 1 is prime if its only positive divisors are 1 and p.
Two choices are made here:
- p is restricted to \mathbb{N} (so p > 0).
- Divisors are restricted to positive divisors.
Both choices keep negatives out. If you accept this definition, the question "can negative numbers be prime?" has a one-word answer: no. The primes are 2, 3, 5, 7, 11, 13, \ldots — always positive, always greater than 1.
But the reason this definition is chosen is more interesting than the definition itself.
Why we exclude negatives: uniqueness of factorisation
The single most important theorem about primes is the Fundamental Theorem of Arithmetic: every integer n \geq 2 has exactly one factorisation into primes, up to the order of factors.
That is the only factorisation. Now try extending primes to include negatives. Suddenly:
You get many factorisations, differing by flipping signs on an even number of factors. Uniqueness is gone, unless you patch the statement to read "unique up to order and signs." That patch is not difficult, but it adds friction to every theorem that uses unique factorisation — and that is almost every theorem in number theory.
By declaring once and for all that primes are positive and working with positive factorisations, you keep every downstream statement clean.
The deeper story: primes up to "units"
In ring theory — the branch of algebra that generalises arithmetic — the notion of "prime" is refined. A unit is a number with a multiplicative inverse in the ring. In \mathbb{Z}, the units are exactly 1 and -1, because those are the only integers whose reciprocal is also an integer.
An irreducible element is one that cannot be written as a product of two non-units. And two elements are associates if they differ by a unit: a \sim b when a = u \cdot b for some unit u. So in \mathbb{Z}, 7 and -7 are associates (-7 = (-1) \cdot 7), and both are irreducible.
Why: "associate" is the algebraic word for "same up to a sign." In \mathbb{Z}, it just means "differ by a -1 flip."
From the ring-theoretic viewpoint, 7 and -7 are both prime — they are two associates of the same prime element. You pick one representative per associate class to be "the" prime, and the choice of positive is a convention (a very reasonable one, because positives are easier to list).
So the rigorous answer to "can -7 be prime?" is: yes, in the broader sense where primes come in associate classes. But when working in \mathbb{Z}, we fix the convention "prime = positive" and work from there.
In the Gaussian integers, the picture is richer
Move beyond \mathbb{Z} and the question becomes genuinely interesting. The Gaussian integers are \mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}. This ring has four units: 1, -1, i, -i. Every prime element has four associates — for instance, 1 + i is prime, and so are -1 - i, -1 + i, and 1 - i (multiplying 1+i by each of the four units).
5 is not prime in \mathbb{Z}[i], because 5 = (2 + i)(2 - i). But 2 + i and 2 - i are both prime in \mathbb{Z}[i]. Notice |2+i|^2 = 5, so 2+i is a "square root of 5" in a sense — and the fact that 5 splits is a theorem (primes p \equiv 1 \pmod 4 split; primes p \equiv 3 \pmod 4 stay prime; 2 ramifies).
You do not need the details — just the point that "which numbers are prime" genuinely depends on the ring. In \mathbb{Z}, primes are the positive irreducibles. In \mathbb{Z}[i], primes include things like 2 + i that are not ordinary integers at all.
What about divisibility by -p?
Here is a subtle place where negatives behave exactly like positives. The statement "7 divides n" and the statement "-7 divides n" are equivalent — any n divisible by 7 is also divisible by -7 (because n/(-7) = -(n/7), and if n/7 is an integer, so is its negative). So divisibility does not distinguish p from -p. This is part of why picking just one sign for "the prime" loses no information.
Modular arithmetic is the same way: n \equiv a \pmod p is the same condition as n \equiv a \pmod{-p}. The modulus could be either sign.
What people sometimes mean by "negative prime"
Occasionally you will hear physicists or computer scientists talk about "negative primes" — they usually just mean -2, -3, -5, \ldots, the negatives of the ordinary primes. This is a harmless abuse of language: it treats every prime as having two signed forms and picks whichever makes a formula or algorithm cleaner. As long as you remember that in \mathbb{Z}, "prime" in the strict sense means positive, the abuse will not cause confusion.
The one-line answer
In \mathbb{Z}, primes are defined as positive — the convention preserves the clean uniqueness of factorisation. In broader rings, -p is an associate of p and shares its prime status. Both views are correct; which one you use depends on whether you are doing school number theory or algebraic number theory.
So in your Class 9 exam, answer "no, primes are positive." In a college abstract algebra course, answer "yes, -p is a prime up to associates, and that distinction matters." Both answers are right — you are just using different definitions.
Related: Number Theory Basics · Why is 1 Not a Prime Number? · Unique Prime Factorisation — Order Doesn't Matter · Why Number Theory Says "Positive Integer" · Coprime vs Prime