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To check whether one integer is prime, test possible divisors up to its integer square root. To generate every prime below a limit, use a sieve such as SymPy’s primerange. These are different tasks: trial division is a clear standard-library solution for one candidate, while a sieve is designed to produce a bounded list.
What counts as a prime number?
A prime is an integer greater than 1 whose only positive divisors are 1 and itself. That definition makes the boundary cases important: negative numbers, 0, and 1 are not prime. The number 2 is prime; it is also the only even prime.
Check whether one number is prime with Python
For a candidate n, it is enough to test possible divisors up to floor(sqrt(n)). If n had a factor larger than its square root, the paired factor would be smaller. Python’s standard-library math.isqrt gives the exact integer square root without floating-point rounding.
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from math import isqrt
def is_prime(n: int) -> bool:
if n < 2:
return False
if n % 2 == 0:
return n == 2
for divisor in range(3, isqrt(n) + 1, 2):
if n % divisor == 0:
return False
return TrueThe function first rejects values below 2, then handles even numbers directly. For an odd candidate, it checks odd divisors starting at 3 and stops at the integer square root. Skipping even divisors avoids checks that cannot succeed after the even case has been handled.
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For example, is_prime(-1), is_prime(0), and is_prime(1) return False; is_prime(2) returns True; and is_prime(12) returns False because it is even and not 2. The odd prime 13 returns True. The odd composite 45 returns False because 3 divides it; the search does not need to continue beyond its square-root bound.
This is a straightforward approach for learning and checking one moderate-sized integer. It is not a benchmarked performance claim, and trial division is not the best fit for very large candidates.
Python documents math.isqrt as the integer square-root function.
Generate all primes below a limit
If you need a list of primes rather than a yes-or-no answer for one number, use a sieve-based method. SymPy’s primerange(a, b) returns primes in the interval that includes a and excludes b.
from sympy import primerange
primes = list(primerange(1, 30))
print(primes)The result is [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]. The upper endpoint is excluded, so this call returns primes less than 30.
SymPy also documents a dynamically growing Sieve of Eratosthenes. Sieve-based generation is generally faster for producing a range of primes than testing each candidate independently, but it uses more memory because values are stored. Choose it when you need a bounded list and can use the additional dependency and memory.
Choose the approach for your task
| Approach | Best fit | Tradeoff or caveat |
|---|---|---|
Trial division with math.isqrt |
Learning or testing one moderate-sized integer | Simple and uses the standard library, but does not scale like specialized primality routines for very large inputs. |
SymPy primerange or sieve |
Generating primes in a bounded interval | Generally faster for range generation, but uses more memory because values are stored; requires SymPy. |
SymPy isprime |
Convenient library-based checks of individual values | Requires SymPy. For inputs greater than 264, its documented method is a probable-prime test rather than a proof. |
For a small, self-contained program, the standard-library function is often the simplest choice. If you already use SymPy or need to generate many primes, its number-theory tools avoid implementing the sieve yourself. There is no universal numeric cutoff established here; the right choice depends on whether you are testing one candidate or generating a range, whether you can add a dependency, and how much memory you can use.
Use SymPy for a library-based primality check
SymPy provides isprime for checking one value. Its documentation describes deterministic Miller–Rabin tests for smaller inputs and a strong BPSW test for inputs greater than 264. SymPy describes the latter as a probable-prime test and reports no known counterexamples; do not treat that documented result for larger inputs as a mathematical proof.
See SymPy’s isprime, primerange, and sieve documentation for the library APIs and their behavior.
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