The Euclid-Euler theorem is a theorem in number theory relating perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it equals a power of two multiplied by a Mersenne prime, a prime number one less than a power of two, in the specific relationship that Euclid and Leonhard Euler together established, with Euclid proving one direction and Euler proving the other. Whether there are infinitely many Mersenne primes, and therefore infinitely many even perfect numbers, remains an open conjecture, and it is likewise unknown whether even a single odd perfect number exists.
Facts
StatementAn even natural number is perfect if and only if it has the form 2 to the power (p minus 1) multiplied by Mp, where Mp is a Mersenne prime. 2 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Euclid-Euler theorem, Wikipedia
Proved By
Source Euclid-Euler theorem, Wikipedia
Source Euclid-Euler theorem, Wikipedia
Sources
1. Wikipedia: Euclid-Euler theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
It states that an even number is perfect if and only if it has the form 2pā1(2p ā 1), where 2p ā 1 is a prime number.
2. Euclid-Euler theorem, Wikipedia
Statement and examples
The Euclid-Euler theorem states that an even natural number is perfect if and only if it has the form 2p-1Mp, where Mp is a Mersenne prime. The perfect number 6 comes from p = 2 in this way, as 22-1M2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
References
Originally read to the Berlin Academy on February 23, 1747, and published posthumously.
- In Branch: Number Theory, Lead sentence
Proved By: Euclid, Lead
The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.
Proved By: Leonhard Euler, Lead
The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.
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