Home › Math & Conversions › Core Math & Algebra › Perfect Number Calculator Perfect Number Calculator – Perfect, Abundant, or Deficient? Enter a positive integer to check if it is a perfect number and see its proper divisors, divisor sum, and classification as perfect, abundant, or deficient. Optionally search for all perfect numbers in a range. Built for number theory students, teachers, and curious learners who want transparent, step-by-step divisor analysis. Perfect number checker Divisor-based classification Choose a mode, enter your integer(s), and click Calculate . The tool computes proper divisors, their sum, and a clear classification. Mode Single number check Scan a range for perfect numbers Integer n Enter a positive integer. For very large n the computation may take longer; this tool is optimized for classroom-sized examples. Quick examples 6 (perfect) 28 (perfect) 12 (abundant) 8 (deficient) Upper bound N N between 2 and 1,000,000. The algorithm uses an O(√n) divisor search for each integer up to N. Range scan is for exploration and teaching. For very large searches, use theoretical results on perfect numbers or specialized software; this tool intentionally prioritizes readability and deterministic output over extreme performance. Calculate Clear Divisor search up to ⌊√n⌋ with symmetry (d, n/d) for efficiency. Results, divisor lists, and classifications will appear here after you click Calculate . What is a perfect number? A positive integer \(n\) is called a perfect number if it equals the sum of its proper divisors , that is, all positive divisors less than \(n\) itself. Formally, if \(\sigma(n)\) is the sum of all positive divisors of \(n\), then \(n\) is perfect when \[ \sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n. \] The first perfect numbers are \(6, 28, 496, 8128,\dots\). For example, the proper divisors of 28 are \(1, 2, 4, 7, 14\), and \(1 + 2 + 4 + 7 + 14 = 28\). Perfect, abundant, and deficient numbers The same divisor-sum test also gives two related classifications: Perfect : \(\sum_{d \mid n,\, d < n} d = n\). Abundant : \(\sum_{d \mid n,\, d < n} d > n\) (the number has “too much” divisor mass). Deficient : \(\sum_{d \mid n,\, d < n} d < n\) (the divisor sum falls short of the number). Many everyday integers such as 8, 10, and 11 are deficient; abundant numbers like 12 and 18 appear regularly; perfect numbers are extremely rare. Even perfect numbers and Mersenne primes All known perfect numbers are even and have a special form discovered by Euclid and Euler: \[ n = 2^{p-1}\,\bigl(2^p - 1\bigr), \] where \(2^p - 1\) is a Mersenne prime , a prime number of the form \(2^p - 1\). Every time a new Mersenne prime \(2^p - 1\) is found, it generates a new even perfect number. Modern record perfect numbers are discovered using large-scale distributed computing projects. Do odd perfect numbers exist? No one knows. Despite centuries of work, no odd perfect number has ever been found . At the same time, there is no proof that odd perfect numbers cannot exist; it remains a deep open problem in number theory. How the calculator finds proper divisors For a given \(n\), the calculator performs an efficient divisor search: It always includes 1 as a proper divisor (for \(n > 1\)). It loops over integers \(d\) from 2 up to \(\lfloor\sqrt{n}\rfloor\). Whenever \(d\) divides \(n\), it adds both \(d\) and \(n/d\) (when distinct) to the divisor list. It removes \(n\) itself from the list, sorts the remaining divisors, and sums them. This algorithm runs in \(O(\sqrt{n})\) time, which is efficient for the typical ranges used in teaching and exploratory work. Perfect number calculator – FAQ Is 1 a perfect number? + No. The proper divisors of 1 are usually taken to be the empty set, so their sum is 0, which is not equal to 1. For this reason, 1 is classified as deficient, not perfect. The smallest perfect number is 6. Why does the range feature stop at 1,000,000? + The limit keeps the tool fast and responsive in a standard web browser while still covering all classic textbook examples. The divisor search is deliberately implemented in a clear, deterministic way rather than using extremely low-level optimizations. For very large searches, dedicated number theory software or distributed projects are more appropriate. Can this tool prove that a number is not perfect? + For any specific integer n in the supported range, the tool computes all proper divisors and their sum. If the sum does not equal n, then n is not perfect—this is a direct check of the definition, not a heuristic. However, proving that no perfect numbers exist beyond a certain range is a much deeper theoretical question and cannot be settled by finite computation alone. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator Perfect Number Calculator (you are here) Chinese Remainder Theorem Set Theory Basics Number theory & probability Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This perfect number calculator is ideal for teaching, exam preparation, and exploratory number theory. For research-level work on perfect numbers or Mersenne primes, consult specialized literature and software that implements advanced primality tests and large-integer arithmetic.

Subcategories in Home › Math & Conversions › Core Math & Algebra › Perfect Number Calculator Perfect Number Calculator – Perfect, Abundant, or Deficient? Enter a positive integer to check if it is a perfect number and see its proper divisors, divisor sum, and classification as perfect, abundant, or deficient. Optionally search for all perfect numbers in a range. Built for number theory students, teachers, and curious learners who want transparent, step-by-step divisor analysis. Perfect number checker Divisor-based classification Choose a mode, enter your integer(s), and click Calculate . The tool computes proper divisors, their sum, and a clear classification. Mode Single number check Scan a range for perfect numbers Integer n Enter a positive integer. For very large n the computation may take longer; this tool is optimized for classroom-sized examples. Quick examples 6 (perfect) 28 (perfect) 12 (abundant) 8 (deficient) Upper bound N N between 2 and 1,000,000. The algorithm uses an O(√n) divisor search for each integer up to N. Range scan is for exploration and teaching. For very large searches, use theoretical results on perfect numbers or specialized software; this tool intentionally prioritizes readability and deterministic output over extreme performance. Calculate Clear Divisor search up to ⌊√n⌋ with symmetry (d, n/d) for efficiency. Results, divisor lists, and classifications will appear here after you click Calculate . What is a perfect number? A positive integer \(n\) is called a perfect number if it equals the sum of its proper divisors , that is, all positive divisors less than \(n\) itself. Formally, if \(\sigma(n)\) is the sum of all positive divisors of \(n\), then \(n\) is perfect when \[ \sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n. \] The first perfect numbers are \(6, 28, 496, 8128,\dots\). For example, the proper divisors of 28 are \(1, 2, 4, 7, 14\), and \(1 + 2 + 4 + 7 + 14 = 28\). Perfect, abundant, and deficient numbers The same divisor-sum test also gives two related classifications: Perfect : \(\sum_{d \mid n,\, d < n} d = n\). Abundant : \(\sum_{d \mid n,\, d < n} d > n\) (the number has “too much” divisor mass). Deficient : \(\sum_{d \mid n,\, d < n} d < n\) (the divisor sum falls short of the number). Many everyday integers such as 8, 10, and 11 are deficient; abundant numbers like 12 and 18 appear regularly; perfect numbers are extremely rare. Even perfect numbers and Mersenne primes All known perfect numbers are even and have a special form discovered by Euclid and Euler: \[ n = 2^{p-1}\,\bigl(2^p - 1\bigr), \] where \(2^p - 1\) is a Mersenne prime , a prime number of the form \(2^p - 1\). Every time a new Mersenne prime \(2^p - 1\) is found, it generates a new even perfect number. Modern record perfect numbers are discovered using large-scale distributed computing projects. Do odd perfect numbers exist? No one knows. Despite centuries of work, no odd perfect number has ever been found . At the same time, there is no proof that odd perfect numbers cannot exist; it remains a deep open problem in number theory. How the calculator finds proper divisors For a given \(n\), the calculator performs an efficient divisor search: It always includes 1 as a proper divisor (for \(n > 1\)). It loops over integers \(d\) from 2 up to \(\lfloor\sqrt{n}\rfloor\). Whenever \(d\) divides \(n\), it adds both \(d\) and \(n/d\) (when distinct) to the divisor list. It removes \(n\) itself from the list, sorts the remaining divisors, and sums them. This algorithm runs in \(O(\sqrt{n})\) time, which is efficient for the typical ranges used in teaching and exploratory work. Perfect number calculator – FAQ Is 1 a perfect number? + No. The proper divisors of 1 are usually taken to be the empty set, so their sum is 0, which is not equal to 1. For this reason, 1 is classified as deficient, not perfect. The smallest perfect number is 6. Why does the range feature stop at 1,000,000? + The limit keeps the tool fast and responsive in a standard web browser while still covering all classic textbook examples. The divisor search is deliberately implemented in a clear, deterministic way rather than using extremely low-level optimizations. For very large searches, dedicated number theory software or distributed projects are more appropriate. Can this tool prove that a number is not perfect? + For any specific integer n in the supported range, the tool computes all proper divisors and their sum. If the sum does not equal n, then n is not perfect—this is a direct check of the definition, not a heuristic. However, proving that no perfect numbers exist beyond a certain range is a much deeper theoretical question and cannot be settled by finite computation alone. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator Perfect Number Calculator (you are here) Chinese Remainder Theorem Set Theory Basics Number theory & probability Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This perfect number calculator is ideal for teaching, exam preparation, and exploratory number theory. For research-level work on perfect numbers or Mersenne primes, consult specialized literature and software that implements advanced primality tests and large-integer arithmetic..