Perfect Numbers

Number Theory · Axiom Academy

Numbers that equal the sum of their proper divisors 1. Definition of Perfect Numbers A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors except the number itself). Let's visualize how 6 and 28 are perfect numbers: 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 2. Euclid's Theorem on Even Perfect Numbers Euclid discovered a remarkable formula that generates all even perfect numbers, proven around 300 BCE in his Elements . This formula connects perfect numbers to a special class of primes. Let's see how it generates the first few perfect numbers: p = 2: 2 1 (2 2 - 1) = 2 × 3 = 6 p = 3: 2 2 (2 3 - 1) = 4 × 7 = 28 p = 5: 2 4 (2 5 - 1) = 16 × 31 = 496 p = 7: 2 6 (2 7 - 1) = 64 × 127 = 8128 3. Connection to Mersenne Primes Primes of the form 2 p - 1 are called Mersenne primes , named after French monk Marin Mersenne (1588-1648). Note: Not every 2 p - 1 is prime, even when p is prime. For example, 2 11 - 1 = 2047 = 23 × 89 is composite. Known Mersenne Primes: As of 2024, only 51 Mersenne primes are known. The largest known perfect number has over 49 million digits! 4. The Mystery of Odd Perfect Numbers One of the oldest unsolved problems in mathematics asks: Do odd perfect numbers exist? This tells us all even perfect numbers follow Euclid's pattern. But what about odd perfect numbers? Have at least 101 prime factors Have at least 10 distinct prime factors

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