I've added a slightly expanded version of the result below to the web
site at
http://www.unsolvedproblems.org/UP/index.htm

Comments welcome!

Tim

--- In UnsolvedProblems@yahoogroups.com, "Tim Roberts" <tsr21@...>
wrote:
>
> It has been known since the time of Euler that an odd perfect
number
> N (if it exists) must have the form N = p^a * Q^2 where p is prime
> and p = a = 1 mod 4 (see for example Dickson 2005). Further, it
has
> been shown that N must equal 1 mod 12, or 9 mod 36 (Touchard 1953,
> Holdener 2002). However, we can do a little better than this.
>
> From either result it is immediately evident that if 3 divides N,
> then 3^k divides N, where k = 0 mod 2.
>
> If k = 0, then N must be of the form 1 mod 12.
>
> If k = 2, then N must be of the form 9 mod 36. Further, since N is
> perfect, we know that 3^0+3^1+3^2 = 1 + 3 + 9 = 13 must divide 2N,
> and hence N = 0 mod 13. Thus, N must satisfy both N = 9 mod 36 and
N
> = 0 mod 13. From the Chinese Remainder Theorem, we can deduce that
N
> must equal 117 mod 468.
>
> If k > 2, then N is divisible by 3^4 = 81. Thus, N must satisfy
both
> N = 9 mod 36 and N = 0 mod 81. From the Chinese Remainder Theorem,
> we can deduce that N must equal 81 mod 324.
>
> Thus, if N is an odd perfect number, it must be of the form
> N = 1 mod 12
> or N = 117 mod 468
> or N = 81 mod 324.
>
> Of course, it is possible to further refine the last of these
results
> in a similar way, by considering separately values of k greater
than
> or equal to 4.
>
>
> References
>
> Dickson, L. E. (2005) History of the Theory of Numbers, Vol. 1:
> Divisibility and Primality. New York: Dover, pp. 3-33.
>
> Holdener J. A. (2002), A theorem of Touchard and the form of odd
> perfect numbers, American Mathematical Monthly, vol. 109, pp. 661-
663.
>
> Touchard J. (1953), On prime numbers and perfect numbers, Scripta
> Mathematica, vol. 19, pp. 35-39.
>