A positive integer n is said to be perfect if
sigma(n) = 2n
where sigma(x) is the classical sum of divisors of the positive integer x.
For example, 6 and 28 are perfect, since
sigma(6) = 1 + 2 + 3 + 6 = 12 = 2(6)
sigma(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2(28).
Currently, as of October 2023, only 51 perfect numbers are known, and they are all even.
It is currently unknown whether there are any odd perfect numbers.
Euler did derive the possible form of an odd perfect number N, and it is
N = p^k m^2
where p is the special prime satisfying p == k == 1 (mod 4) and gcd(p, m) = 1.
Here is my question:
Do you know of any other papers outlining a GCD approach to odd perfect numbers, apart from the following articles?
- Gcd in Odd perfect number by Devansh Singh (https://www.researchgate.net/publication/363799063)
- A new approach to odd perfect numbers via GCDs by Jose Arnaldo Bebita Dris (https://www.researchgate.net/publication/358655289)
Please advise.
Hello Jose,
unfortunately, I don't know any paper tackling the odd perfect number question via a GCD approach. I'm actually reading the ones you mentioned with your question. Sorry for being of very little help this time around.
Best regards
Roberto
Hello Roberto Violi : That is totally fine! I hope you enjoy reading the second paper.
Best regards,
Arnie Dris
Hello Jose,
It is good to hear from you. The Marymount campus that I worked at did not survive the COVID pandemic and I am currently retired. Over the last few years, I have been working on results concerning the generalized Fermat equation. I am not aware of other papers that study the GCD approach to odd perfect numbers. Sorry I couldn't be more helpful.
Best wishes,
Richard Ryan
Hello Richard Ryan ,
I am so glad to learn that you are fine and doing well.
I do am aware that this GCD approach to odd perfect numbers could be extended to a GCD-LCM approach, as we currently know that the value of
gcd(m^2, sigma(m^2))
is
gcd(m^2, sigma(m^2)) = sigma(m^2)/p^k = m^2 / (sigma(p^k)/2).
For instance, I am getting that
gcd(m^2, sigma(m^2)) lcm(m^2, sigma(m^2)) = m^2 sigma(m^2)
so that
lcm(m^2, sigma(m^2)) = p^k m^2.
Lastly, I also obtain
lcm(m, sigma(m^2))
= [ m sigma(m^2)) sigma(p^k) ] / [ (2m) gcd(sigma(p^k)/2, m) ]
= p^k m^2 / gcd(sigma(p^k)/2, m)
and
lcm(sigma(p^k), sigma(m^2))
= [ 2p^k m^2 (sigma(p^k)/2) ] / [ gcd(sigma(p^k)/2, m) ]^2
= p^k m^2 sigma(p^k) / [ gcd(sigma(p^k)/2, m) ]^2
= lcm(m, sigma(m^2)) [sigma(p^k) / gcd(sigma(p^k)/2, m)]
= [ lcm(m, sigma(m^2)) 2p^k m^2 ] / [ gcd(sigma(p^k)/2, m) sigma(m^2) ]
= lcm(m, sigma(m^2)) [ p^k m^2 / gcd(sigma(p^k)/2, m) ] [ 2/sigma(m^2) ]
= [ lcm(m, sigma(m^2)) ]^2 [ 2/sigma(m^2) ],
so that
[ sigma(m^2)/2 ] [ sigma(p^k)/sigma(p^k) ] [ lcm(sigma(p^k), sigma(m^2)) ]
= [ lcm(m, sigma(m^2)) ]^2
[ lcm(m^2, sigma(m^2)) lcm(sigma(p^k), sigma(m^2)) ] =
sigma(p^k) [ lcm(m, sigma(m^2)) ]^2.
Compare the last equation with
G x H = I^2
where
G = gcd(sigma(p^k), sigma(m^2)) = [ gcd(sigma(p^k)/2, m) ]^2 / [sigma(p^k)/2]
H = gcd(m^2, sigma(m^2)) = sigma(m^2)/p^k = 2m^2/sigma(p^k)
I = gcd(m, sigma(m^2)) = (m/(sigma(p^k)/2)) gcd(sigma(p^k)/2, m).
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