Numeri perfetti

Perfect numbers are the oldest subject in number theory that still has an open problem attached to it. Euclid's Elements, compiled around 300 BCE, closes Book IX with Proposition 36: if the running sum 1 + 2 + 4 + … + 2^(p−1) happens to be prime, then multiplying that sum by its last term produces a perfect number. In modern notation, 2^(p−1)(2^p − 1) is perfect whenever 2^p − 1 is prime. The first four values — 6, 28, 496 and 8128 — were already in circulation.

Around 100 CE Nicomachus of Gerasa listed those four in his Introduction to Arithmetic and wrapped them in two confident generalisations: that there is exactly one perfect number of each digit-length, and that they end alternately in 6 and 8. Both are false. The fifth perfect number has eight digits rather than five, and 33550336 follows 8128 without the alternation surviving. The claims nevertheless went unchallenged for more than a millennium, largely because nobody produced a fifth example to test them against. The fifth perfect number, 33550336, finally surfaces in an anonymous Latin manuscript of the mid-fifteenth century, and Pietro Antonio Cataldi worked out the sixth and seventh — 8589869056 and 137438691328 — in 1588.

Ibn al-Haytham, writing in Cairo around 1000 CE, was the first to suspect that Euclid's construction catches every even perfect number, and he attempted a proof. The proof came from Leonhard Euler in the 1740s, which is why the result is now called the Euclid–Euler theorem: the even perfect numbers and the Mersenne primes are the same subject in two notations.

What Euler did not settle is the odd case. No odd perfect number has been found and none has been ruled out. The search for new perfect numbers is therefore a search for Mersenne primes, run since 1996 by the volunteer Great Internet Mersenne Prime Search; in October 2024 Luke Durant found 2^136279841 − 1 using a network of cloud GPUs, making the fifty-second known perfect number a figure of roughly 82 million digits.

Proprietà principali

• A perfect number equals the sum of its proper divisors: 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14. Equivalently σ(n) = 2n.
• Euclid–Euler theorem: an even number is perfect if and only if it has the form 2^(p−1)(2^p − 1) with 2^p − 1 prime, so even perfect numbers correspond exactly to Mersenne primes.
• Fifty-two perfect numbers are known as of October 2024, the largest built from the Mersenne prime 2^136279841 − 1.
• Every even perfect number is triangular: 2^(p−1)(2^p − 1) is the sum of the integers from 1 up to 2^p − 1.
• Every even perfect number except 6 is a sum of consecutive odd cubes: 28 = 1³ + 3³, and 496 = 1³ + 3³ + 5³ + 7³.
• In binary, an even perfect number is p ones followed by p − 1 zeros: 28 is 11100 and 496 is 111110000.
• Every even perfect number ends in the digit 6 or in the digits 28.
• No odd perfect number is known. If one exists it must exceed 10^1500 and have at least ten distinct prime factors.

Dove si incontrano

• Euclid’s Elements Book IX, Proposition 36 is the oldest surviving theorem about perfect numbers, and it remains the working construction for the even case.
• The Great Internet Mersenne Prime Search has been a volunteer distributed-computing project since 1996; every perfect number discovered since then has come out of it, and the Electronic Frontier Foundation has paid cash awards for record-size primes.
• The Lucas–Lehmer test and the large fast-Fourier-transform multiplication it needs are what Mersenne searching actually exercises, which is why the search software doubles as a CPU stress test for overclockers and hardware reviewers.
• Augustine of Hippo argued in The City of God that creation took six days because six is perfect, rather than six being perfect because of creation — a theological reading of the arithmetic, not a mathematical result.
• Greek and later writers attached significance to 6 and especially to 28, often tying the latter to the roughly 28-day lunar month. This is a numerological tradition rather than an established connection.