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Odd perfect numbers

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It is not known whether odd perfect numbers exist or not! Mathematicians have been able to prove all sorts of necessary (but not sufficient) requirements for the existence of such numbers without being able to prove either that they do exist or that they don't exist.

Search for odd perfect numbers

For an odd number
n
to be perfect, we must have
where
k, k > 1,
is an odd factor and
pa
is an odd prime power factor which is coprime to
k
, such that

or

where
σ (n)
is the sum of divisors of
n
. One way to search for odd perfect numbers is to consider, for each deficient odd
k, k > 1,
if some odd prime power
pa
which is coprime to
k
, when multiplied by
k
yields an odd perfect number. (Since all positive multiples of abundant numbers are also abundant, there is no point in considering abundant odd
k
 's.)

One might consider the simpler case

where
k, k > 1,
is an odd factor and
p
is an odd prime factor which does not divide
k
, such that
where
σ (n)
is the sum of divisors of
n
.

Equivalently, we must have

thus we need to find an odd
k, k > 1,
such that
where
p
happens to be an odd prime.
A008438 Sum of divisors of
2n  +  1, n   ≥   0
.
{1, 4, 6, 8, 13, 12, 14, 24, 18, 20, 32, 24, 31, 40, 30, 32, 48, 48, 38, 56, 42, 44, 78, 48, 57, 72, 54, 72, 80, 60, 62, 104, 84, 68, 96, 72, 74, 124, 96, 80, 121, 84, 108, 120, ...}

Odd spoof perfect numbers

Main article page: Descartes number

In 1638, Descartes found the following "odd spoof perfect number" (no other has ever been found!):

that is odd and perfect only if you suppose (incorrectly) that

is a "spoof-prime factor," giving the "spoof prime factorization"

for which the "freestyle sum of divisors" (i.e. the sum of divisors function where one is free to consider some composite factors as "spoof-prime factors") yields

References

  • Eric W. Weisstein, CRC Encyclopedia of Mathematics, Volume II, CRC Press (2009), p. 2730.

External links