Use trial division to check each number from the lower bound through the upper bound, inclusive. The program below skips values below 2 and tests possible divisors only up to the candidate’s integer square root.
Python program for an inclusive range
This version treats both low and high as included endpoints. Set the two values near the bottom of the code to choose the interval.
import math
def is_prime(n):
if n < 2:
return False
for divisor in range(2, math.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 = 1
high = 50
print(primes_in_range(low, high))For the example bounds, the output is:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]The list contains primes from 1 through 50; neither endpoint is prime in this example. The helper also handles intervals below 2, and a reversed interval produces an empty list.
How the primality check works
Exclude values below 2
A prime is an integer greater than 1 whose only positive divisors are 1 and itself. Negative numbers, 0, and 1 are therefore not prime. The early if n < 2 check handles them directly.
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Check divisors only through the square root
The expression n % divisor == 0 detects an even division, so n is composite when it is true for any candidate divisor. The loop only needs to reach the square root: if a number has a factor above its square root, its paired factor is below it. Python’s math.isqrt returns the floor of the exact square root for a nonnegative integer and has been available since Python 3.8.
The + 1 in range(2, math.isqrt(n) + 1) matters because Python’s range excludes its stop value. It ensures the integer square root itself is checked. For example, 25 must be tested for divisibility by 5; omitting that endpoint would incorrectly classify it as prime. If no divisor is found, the function returns True.
Adjusting the bounds and output
Python’s range includes its starting value but excludes its stopping value. The outer loop therefore uses high + 1 to include the requested upper endpoint. To use a half-open interval [low, high) instead, change the list comprehension to range(low, high) and describe the bounds accordingly.
The function returns a list, which is convenient if the calling code needs to use the primes later. To print one prime per line instead, replace the final call with:
for prime in primes_in_range(low, high):
print(prime)When to use a sieve instead
The helper above performs a separate divisibility check for each candidate. That makes it a clear choice for a beginner exercise or for checking a modest interval. If the task is to generate every prime from 2 up to a large limit, a Sieve of Eratosthenes may fit better: mark multiples of each prime as composite, beginning at that prime’s square. The NIST Dictionary of Algorithms and Data Structures notes that the basic sieve uses Θ(N) memory; segmented sieves reduce memory needs. There is no universal crossover point—the better method depends on the input and implementation.
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