Number theory

• How to Check Whether a Number Is Perfect
  A positive integer is perfect when its proper divisors sum to itself. Learn a direct divisor-pair check and a faster formula using prime factors.
• Find Positive Integers with GCD 10 and LCM 100
  Write each number as 10 times a coprime factor; the factor pairs of 10 give all solutions: (10, 100), (20, 50), and their reversals.
• Examples of Open Problems Solved by Short Proofs
  See examples where a compact argument settled a mathematical problem, and learn why a short proof can hide a long search or earlier work.
• What Are the Conditions for Eisenstein’s Criterion?
  Eisenstein’s criterion requires a prime to divide every nonleading coefficient but not the leading coefficient or the constant term squared.
• How Do You Compute Irrational Numbers?
  Irrational numbers are computed by algorithms that produce rational approximations to a chosen precision, often with an error bound.
• What Does ω(n) Mean? Distinct Prime Factors Explained
  The arithmetic function ω(n) counts the distinct prime divisors of n. Learn its definition, value at 1, difference from Ω(n), and how to compute it.
• What Does the Height of an Algebraic Equation Mean?
  The height of an algebraic equation usually measures coefficient size, but algebraic-number height is a distinct invariant; here are both definitions.