First perfect number
- Input
- 6
- Expected output
- 1 + 2 + 3 = 6
Sum of the proper divisors equals the number.
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what is a perfect number
A perfect number is one that equals the sum of its proper divisors, meaning all its divisors except itself. The smallest is 6, since 1 + 2 + 3 = 6. The next is 28, then 496 and 8,128. Type any of them into the calculator and the perfect tag appears next to the sum.
Sum of the proper divisors equals the number.
28 = 4 × 7, with p = 3.
496 = 16 × 31, with p = 5.
16 is larger than 12.
9 is smaller than 15.
975 is larger than 945; below 945 every odd number is deficient.
There are two ways. The direct one is to test the whole numbers from 1 up to the square root of the number: each time the division is exact you gain two divisors, the tested one and the quotient. For 36, testing 1, 2, 3, 4, 5 and 6 gives the pairs (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6). Factoring is faster for large numbers: break the number into primes and combine the powers, which is what the Step by step tab shows.
The 6. Its proper divisors are 1, 2 and 3, and 1 + 2 + 3 = 6.
None has been found so far and nobody has proved none exists. It is an open problem. Every known perfect number is even and follows the Euclid-Euler formula.
No. The 1 has no proper divisors, so the sum is 0, smaller than 1. It is classified as deficient.
The divisors include the number itself: those of 28 are 1, 2, 4, 7, 14 and 28. The proper ones leave 28 out. A number is perfect when the sum of all its divisors is exactly twice the number: 56 = 2 × 28.
36 has 9 divisors
The proper divisors add up to 55, which is 19 more than 36: an abundant number.
Divisors that are prime are highlighted.
Accepts whole numbers of up to 18 digits. Everything runs in your browser with exact arithmetic.
Related: GCD and LCM · Prime numbers · Simplify fractions