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Outbyte PC Repair FREERepair Windows errors before they cause bigger problemsFix Now →Outbyte Driver Updater FREEScan for outdated or missing drivers - takes under a minuteDriver Scan →Use trial division to check each number in the interval: skip values below 2, then test possible divisors only up to the number’s integer square root. The program below accepts an inclusive lower and upper bound and returns every prime between them.
Python program for an inclusive range
This version includes both endpoints. For example, if you enter 2 and 10, it returns [2, 3, 5, 7].
from math import isqrt
def is_prime(n):
if n < 2:
return False
for divisor in range(2, isqrt(n) + 1):
if n % divisor == 0:
return False
return True
def primes_in_range(low, high):
return [n for n in range(low, high + 1) if is_prime(n)]
low = int(input("Enter the lower bound: "))
high = int(input("Enter the upper bound: "))
print(primes_in_range(low, high))range(low, high + 1) includes high, because Python’s range stops before its stop value. If low is greater than high, the loop has no candidates and the result is an empty list.
How the prime check works
Exclude values below 2
A prime is an integer greater than 1 whose only positive divisors are 1 and itself. That means negative numbers, 0, and 1 are not prime; the first check handles all of them.
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Test divisibility with the remainder operator
The expression n % divisor == 0 means that divisor divides n evenly. When that happens, n is composite, so is_prime returns False immediately. If no divisor is found, it returns True.
Stop at the integer square root
There is no need to test every number below n. If a composite number can be written as a product of two factors, at least one factor must be less than or equal to its square root; otherwise their product would be greater than n. Checking through the square root therefore finds any possible factor.
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math.isqrt(n) returns the floor of the exact square root for a nonnegative integer. It avoids using a floating-point square root to set the divisor bound and is available in Python 3.8 and later. The + 1 in range(2, isqrt(n) + 1) makes the loop include that bound, which matters for perfect squares such as 9 and 25.
Example output
For the inclusive interval from 1 to 50, the program prints:
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[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]Values below 2 are omitted, and 50 is checked but is not prime.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.When to use a sieve instead
The helper-function approach checks candidates one by one and is easy to follow when an exercise asks whether numbers in a range are prime. If the task is to generate all primes up to a bound, a Sieve of Eratosthenes is another approach: start with integers from 2 through the bound, then mark multiples of each prime as composite. Marking can start at the prime’s square, because smaller multiples have already been marked by smaller prime factors.
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| Approach | Best fit | Memory consideration |
|---|---|---|
| Trial division | Checking one number or iterating over a modest interval; straightforward to explain with an is_prime helper. |
Keeps little state beyond the current candidate and divisor. |
| Basic Sieve of Eratosthenes | Generating every prime up to a limit by eliminating composite multiples together. | Uses memory proportional to the bound; NIST describes the naive implementation as requiring Θ(N) memory. |
| Segmented sieve | Generating primes across a large interval when a basic sieve’s memory use is a concern. | Uses segments to reduce memory needs. |
There is no universal input-size crossover established here; it depends on the range and implementation. Choose trial division for clarity and individual checks, and consider a sieve when the task is bulk generation.
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