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List of perfect numbers - Wikipedia, the free encyclopedia
The following is a list of the known perfect numbers, including the Mersenne prime exponent p which generates them with the expression 2 p −1 ×(2 p − 1) where 2 p − 1 is a Mersenne ...
en.wikipedia.org/wiki/List_of_perfect_numbers |
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(1) The nth perfect number has n digits. (2) All perfect numbers are even. (3) All perfect numbers end in 6 and 8 alternately. (4) Euclid's algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form 2k-1 ... The first error in Mersenne's list was discovered in 1876 by Lucas.
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Let M(p) = 2p-1 and P(p) = 2p-1(2p-1). The list of all known primes p for which M(p) is a Mersenne prime (therefore P(p) is a perfect number) follows: ... Is there an odd perfect number? We know that all even perfect numbers are a Mersenne prime times a power of two (Theorem One above), but what about odd perfect...
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A proof that all even perfect numbers are a power of two times a Mersenne prim ... ; (From the Prime Pages' list of proofs) ... It is still unknown if there are any odd perfect numbers (but if there are, they are large and have many prime factors).
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This may be hard to believe, but there are still more! In fact, there are 35 TOTAL!!! However, it would take too long to write them all out. All EVEN perfect numbers follow this form: ... Click here for a list of all 35 perfect numbers.
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Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n. ... And a further list of multiply perfect numbers sorted only by their factorizations (built from the 5271 MPNs of the master list and gziped 199 kB). It has each line cut down to 120 columns and no discovery information...
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It is conjectured that there are infinitely many amicable pairs, although, as for perfect numbers, this is not known. Here is a list of all the amicable pairs with lower member below 2.01*1011, and here is a longer (but less annotated) list comprising the first 5001 amicable pairs.
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