[seqfan] Re: Takashi Agoh on Giuga's conjecture

[Max Alekseyev]

The statement in the paper looks weird.
If n is even and p is its odd prime divisors, then n-p is odd and B(n-p) = 0.
So the first congruence cannot hold for even n and odd p.

Maybe the statement holds only for odd Guiga numbers?

Max

On Fri, Aug 3, 2012 at 12:26 PM, Paolo Lava <paoloplava at gmail.com> wrote:
> In the article “On Giuga’s Conjecture”
>
> (pages 506:504 in
> http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN365956996_0087 ),
> Takashi Agoh states that if n is a Giuga number (A007850) then for any
> prime factor p of n we have:
>
> i) pB(n-p)==p-1 (mod p^3)
>
> ii) pB(n/p 1)==p-1 (mod p^2)
>
> iii) pB(((n/p)-1)/p)==p-1 (mod p)
>
> where B(i) is the Bernoulli number of index i.
>
> I tried to compute some number that satisfy the above relations and I got
> (Maple program):
> i) 9, 25, 27, 45, 81, 125, 169, 225, 243, 325, 405, 625, 729, 1125, 2025,
> 2187, 2197, 3125, 3645,...
> ii) 25, 125, 169, 325, 625, 2197, 3125, 4225...
> iii) ?
>
> No Giuga number at all: for sure there must be something I misinterpret...
>
> Paolo
>
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