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perfect number∗
Wkbj79†
2013-03-21 12:24:34
An positive integer n is called perfect if it is the sum of all positive divisors
of n less than n itself. It is not known if there are any odd perfect numbers, but
all even perfect numbers have been classified according to the following lemma:
Lemma 1. An even number is perfect if and only if it equals 2k−1 (2k − 1) for
some integer k > 1 and 2k − 1 is prime.
Proof. Let σ denote the sum of divisors function. Recall that this function is
multiplicative.
Necessity: Let p = 2k − 1 be prime and n = 2k−1 p. We have that
σ(n)
= σ(2k−1 p)
= σ(2k−1 )σ(p)
=
(2k − 1)(p + 1)
=
(2k − 1)2k
=
2n,
which shows that n is perfect.
Sufficiency: Assume n is an even perfect number. Write n = 2k−1 m for some
odd m and some k > 1. Then we have gcd(2k−1 , m) = 1. Thus,
σ(n) = σ(2k−1 m) = σ(2k−1 )σ(m) = (2k − 1)σ(m).
Since n is perfect, σ(n) = 2n by definition. Therefore, σ(n) = 2n = 2k m.
Piecing together the two formulas for σ(n) yields
2k m = (2k − 1)σ(m).
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Thus, (2k − 1) | 2k m, which forces (2k − 1) | m. Write m = (2k − 1)M . Note
that 1 ≤ M < m. From above, we have:
2k m
=
(2k − 1)σ(m)
2 (2 − 1)M
=
(2k − 1)σ(m)
k
k
2k M
= σ(m)
Since m | m by definition of divides and M | m by assumption, we have
2k M = σ(m) ≥ m + M = 2k M,
which forces σ(m) = m + M . Therefore, m has only two positive divisors, m
and M . Hence, m must be prime, M = 1, and m = (2k − 1)M = 2k − 1, from
which the result follows.
The lemma can be used to produce examples of (even) perfect numbers:
• If k = 2, then 2k − 1 = 22 − 1 = 3, which is prime. According to the
lemma, 2k−1 (2k − 1) = 22−1 · 3 = 6 is perfect. Indeed, 1 + 2 + 3 = 6.
• If k = 3, then 2k −1 = 23 −1 = 7, which is prime. According to the lemma,
2k−1 (2k − 1) = 23−1 · 7 = 28 is perfect. Indeed, 1 + 2 + 4 + 7 + 14 = 28.
• If k = 5, then 2k − 1 = 25 − 1 = 31, which is prime. According to the
lemma, 2k−1 (2k − 1) = 25−1 · 31 = 496 is perfect. Indeed, 1 + 2 + 4 + 8 +
16 + 31 + 62 + 124 + 248 = 496.
Note that k = 4 yields that 2k − 1 = 24 − 1 = 15, which is not prime.
The sequence of known perfect numbers appears in the OEIS as sequence
A000396.
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