Perfect Number Calculator: Divisor Sum & Euclid Formula
Check if a number is perfect, deficient or abundant using the sum of proper divisors. Discover Euclid's formula and Mersenne prime connections.
Input Number & Search Range
28 is a Perfect Number!
The aliquot sum of proper divisors strictly equals 28. It satisfies Euclid's classical criterion: s(n) = n.
Target Number (n)
Aliquot Sum s(n)
Abundancy Index I(n)
Proper Divisors Count
Number vs Divisors Sum
Ratio: 100.0%Equal bar lengths indicate a perfect number. Divisor sum exceeding target indicates abundance.
Proper Divisors Contribution
5 proper divisorsDistribution of the largest proper divisors making up the aliquot sum.
Perfect Numbers in Range (2 to 1000)
Complete Divisor Decomposition & Factor Pairs
Total Divisors Sum: 28| # | Proper Divisor | Complement Factor | Cumulative Aliquot Sum |
|---|---|---|---|
| 1 | 1 | 1 × 28 = 28 | 1 |
| 2 | 2 | 2 × 14 = 28 | 3 |
| 3 | 4 | 4 × 7 = 28 | 7 |
| 4 | 7 | 7 × 4 = 28 | 14 |
| 5 | 14 | 14 × 2 = 28 | 28 |
Euclid-Euler Theorem: Even Perfect Numbers & Mersenne Primes
Every even perfect number takes the form 2^(p-1) × (2^p - 1) where 2^p - 1 is prime.
| Prime p | Mersenne Prime (2^p - 1) | Euclid Formula | Perfect Number | Digits | Discovered By |
|---|---|---|---|---|---|
| p = 2 | 3 | 2¹ × 3 | 6 | 1 | Ancient Greeks (Pythagoreans) |
| p = 3 | 7 | 2² × 7 | 28 Match | 2 | Ancient Greeks |
| p = 5 | 31 | 2⁴ × 31 | 496 | 3 | Ancient Greeks |
| p = 7 | 127 | 2⁶ × 127 | 8,128 | 4 | Nicomachus (~100 CE) |
| p = 13 | 8,191 | 2¹² × 8,191 | 33,550,336 | 8 | Medieval manuscript (~1456) |
| p = 17 | 131,071 | 2¹⁶ × 131,071 | 8,589,869,056 | 10 | Pietro Cataldi (1588) |
| p = 19 | 524,287 | 2¹⁸ × 524,287 | 137,438,691,328 | 12 | Pietro Cataldi (1588) |
| p = 31 | 2,147,483,647 | 2³⁰ × (2³¹ - 1) | 2,305,843,009,213,693,952 | 19 | Leonhard Euler (1772) |
Variables Explained
n
The positive integer under inspection (tested for perfection, abundance, or deficiency).
s(n) (Aliquot Sum)
The sum of all proper divisors of n, excluding the number n itself.
σ(n) (Divisor Function)
The sum of all positive divisors of n, including n (σ(n) = s(n) + n).
Abundancy Index I(n)
The ratio σ(n) / n. If I(n) = 2, n is perfect; if I(n) < 2, n is deficient; if I(n) > 2, n is abundant.
M_p = 2^p - 1
The Mersenne number for prime exponent p. When M_p is prime, 2^(p-1) × M_p generates an even perfect number.
Important Notes
Exact Integer Verification: This calculator uses deterministic trial factorization to compute exact proper divisors for numbers up to 100,000,000. Range searches above 10,000 are capped to preserve browser responsiveness.
Theoretical Mathematics: All discovered perfect numbers to date are even and generated by Mersenne primes. The existence of odd perfect numbers remains one of the oldest unsolved open problems in mathematics.
Frequently Asked Questions
Related Calculators
Quick Tips
- 1Double-check your input values for accuracy
- 2Use the correct units of measurement
- 3Round results to appropriate decimal places
- 4Consider the context when interpreting results