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The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form 2 p−1 M p, where M p is a Mersenne prime. [1] The perfect number 6 comes from p = 2 in this way, as 2 2−1 M 2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
converges to the finite value 2, and there are consequently more primes than squares. This proves Euclid's Theorem. [10] Symbol used by Euler to denote infinity. In the same paper (Theorem 19) Euler in fact used the above equality to prove a much stronger theorem that was unknown before him, namely that the series
In 1747, Leonhard Euler completed what is now called the Euclid–Euler theorem, showing that these are the only even perfect numbers. As a result, there is a one-to-one correspondence between Mersenne primes and even perfect numbers, so a list of one can be converted into a list of the other. [1] [5] [6]
Euclid–Euler theorem (number theory) Euler's partition theorem (number theory) Euler's polyhedron theorem ; Euler's quadrilateral theorem ; Euler's rotation theorem ; Euler's theorem (differential geometry) Euler's theorem (number theory) Euler's theorem in geometry (triangle geometry) Euler's theorem on homogeneous functions (multivariate ...
In the 4th century BC, Euclid proved that if 2 p − 1 is prime, then 2 p − 1 (2 p − 1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. [5] This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.
Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. [2] This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.
Download as PDF; Printable version; In other projects Wikidata item; ... Erdős–Tetali theorem; Euclid–Euler theorem; Euler's theorem; F. Faltings' product theorem;
Any theorem in Euclidean geometry; Any theorem in Euclid's Elements, and in particular: Euclid's theorem that there are infinitely many prime numbers; Euclid's lemma, also called Euclid's first theorem, on the prime factors of products; The Euclid–Euler theorem characterizing the even perfect numbers; Geometric mean theorem about right ...