Do these 3 things before closing this tab:
1Repair Windows errors before they cause bigger problems2Fix the driver behind crashes, sound loss and screen glitches3Clear out junk files and repair common Windows errorsA perfect number equals the sum of its positive divisors other than itself. In Python, test that definition by adding divisors that divide the number evenly with the integer remainder operator %, then compare the sum with the number.
What is a perfect number?
A perfect number is equal to the sum of its proper divisors: its positive divisors excluding the number itself. Euclid’s Elements, Book VII, Definition 22, describes a perfect number as “that which is equal to the sum its own parts.”
For example, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6. The proper divisors of 28 are 1, 2, 4, 7, and 14; their sum is 28. The first four perfect numbers are 6, 28, 496, and 8128 (Euclid’s Elements, Book VII, Definition 22).
Beginner Python program to test one number
This function checks each possible proper divisor from 1 up to, but not including, the number. The modulo operator % gives the remainder; a remainder of zero means the divisor divides evenly.
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def is_perfect(n):
if n <= 0:
return False
divisor_sum = 0
for divisor in range(1, n):
if n % divisor == 0:
divisor_sum += divisor
return divisor_sum == n
number = 28
if is_perfect(number):
print(number, "is perfect")
else:
print(number, "is not perfect")The function rejects non-positive inputs because the definition here concerns positive divisors of a positive integer. For 1, the loop has no values and the proper-divisor sum is 0, so it correctly returns False. Use % rather than / for the divisibility test: Python’s / operator returns a floating-point result, while the remainder test directly determines whether division is exact. Indentation matters because it groups the statements inside the function, loop, and condition (Python tutorial: An informal introduction to Python).
List perfect numbers below a limit
To search a range, call the test for each candidate. This example treats limit as exclusive: it checks numbers from 1 through limit - 1, not the limit itself.
def perfect_numbers_below(limit):
results = []
for candidate in range(1, limit):
if is_perfect(candidate):
results.append(candidate)
return results
print(perfect_numbers_below(10_000))Expected output:
[6, 28, 496, 8128]The exercise of listing the first four perfect numbers appears in a Python teaching manual (Python programming exercises). Searching below 10,000 finds exactly these four.
Check the result by hand
Verifying both positive and negative cases helps catch errors such as including the number itself in the divisor sum.
- 6: proper divisors are 1, 2, and 3; their sum is 6, so the function returns
True. - 28: proper divisors are 1, 2, 4, 7, and 14; their sum is 28, so the function returns
True. - 12: proper divisors are 1, 2, 3, 4, and 6; their sum is 16, so the function returns
False.
Optional optimization: check divisor pairs
The full scan is easiest to understand, but for larger candidates you can check only as far as the integer square root. Divisors come in pairs: if d divides n, then n // d is the paired divisor. For example, for 28, checking 2 reveals the pair 14.
from math import isqrt
def is_perfect_fast(n):
if n <= 1:
return False
divisor_sum = 1
for divisor in range(2, isqrt(n) + 1):
if n % divisor == 0:
paired_divisor = n // divisor
divisor_sum += divisor
if paired_divisor != divisor:
divisor_sum += paired_divisor
return divisor_sum == nThe initial sum of 1 accounts for 1 as a proper divisor of every candidate greater than 1. The equality check on the pair prevents counting a square root twice: when n is a square and divisor equals its paired quotient, add it only once. This approach requires fewer divisibility checks as the candidate grows, but no measured timing comparison is implied; the straightforward scan is usually clearer for a first exercise.
Why the first four numbers have this pattern
There is also a number-theory characterization for even perfect numbers: if 2^n - 1 is prime, then 2^(n - 1)(2^n - 1) is an even perfect number. For instance, using n = 3 gives 28 because 2^3 - 1 = 7 is prime. This is a useful explanation for a pattern, not a replacement for the program’s divisor-sum test (Number Theory in Context and Interaction).
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