Even Perfect Number
Euclid-Euler form of even perfect numbers.
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Example
6, 28, 496, ...
How it works
This formula gives every even perfect number: a number equal to the sum of all its positive divisors except itself. It works because if p is prime and 2^p - 1 is also prime, then the divisors of N = 2^(p-1)(2^p - 1) combine in just the right way so that their total is exactly 2N, making the proper divisors sum to N. Euclid proved that every number of this form is perfect, and Euler later showed that no other even perfect numbers exist.
When to use it
Use this formula when you need to generate or test even perfect numbers in number theory. It also appears in proofs and exercises involving Mersenne primes, divisor sums, and the classification of perfect numbers.
Worked example, step by step
Take p = 5, since 2^5 - 1 = 31 is prime. N = 2^(p-1)(2^p - 1) N = 2^(5-1)(2^5 - 1) N = 2^4(32 - 1) N = 16 × 31 N = 496 Check the divisor sum: 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 So 496 is an even perfect number.
Common mistakes to avoid
- Using any prime p without checking that 2^p - 1 is also prime
- Forgetting the exponent is p-1, not p
- Thinking the formula generates odd perfect numbers as well
Did You Know?
A perfect number equals the sum of its proper divisors, and only about 50 are known despite millennia of searching.
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