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

[Paolo Lava]

Max,

if the three expressions hold just for odd Giuga numbers then I do not
understand why it was not expressly written in the text of the proposition
(by the way, at the present no odd Giuga number is known).
Even the proof presented by Agoh is a little bit weird to me.

As last chance I could ask Agoh a clarification: it is still in the
teaching staff at Tokyo University of Science  (
www.tus.ac.jp/en/grad/riko_math.html#master)

Paolo


2012/8/3 Max Alekseyev <maxale at gmail.com>

> 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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