Math 2.0

Number theory Definition

Perfect number

Builds on Sum of divisors

Definition

A positive integer nn is perfect if it is equal to the sum of its positive divisors other than nn. Equivalently, σ(n)=2n\sigma(n) = 2n, where σ\sigma is the sum of divisors.

Remarks

The first perfect numbers are 66, 2828, 496496 and 81288128. For example, 28=1+2+4+7+1428 = 1 + 2 + 4 + 7 + 14. The Euclid–Euler theorem describes all even perfect numbers by Mersenne primes. No odd perfect number is known, and it is an open problem whether one exists.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionDivisors of a natural number, definition, no evidence levelDivisors of a natural…definitionDivisor function, definition, no evidence levelDivisor functiondefinitionSum of divisors, definition, no evidence levelSum of divisorsdefinitionPerfect number, definition, no evidence levelPerfect numberdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Arithmetic function definition
  2. Divisibility definition
  3. Divisors of a natural number definition
  4. Divisor function definition
  5. Sum of divisors definition
  6. Perfect number definition

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Perfect_number, lead section
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