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

[Paolo Lava]

Max,

here below is the answer I received from professor Takashi Agoh.

The interesting thing is that he consider Giuga numbers the integers for
which p^2(p-1)| n-p for all prime factors p of n and "weak" Giuga numbers the
integers for which p^2| n-p for all prime factors p of n (A007850). This
sound totally new to me!
***************************************************************************************************

Dear Professor Paolo P. Lava:

Thank you very much for your email in the matter of Proposition 6
in my article.

Unfortunately I do not understand exactly your question, but I will
write below my comments concerning Proposition 6.

(1) The number n is a Giuga number <=> n is composite and p^2(p-1)| n-p
for all prime factors p of n.

(2) The number n is a weak Giuga number <=> n is composite and p^2 | n-p
for all prime factors p of n. For example, n=30, 858, 1722, ....

As you wrote, Proposition 6 asserts that if n is a Giuga number stated
in (1) (not in (2)), then the following congruences hold:

(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).

So the above congruences are just necessary conditions for (1), but
not sufficient conditions. As you computed, there are (infinitely)
many n's satisfying, for example, congruence (i), but almost of all
these numbers you listed are not Giuga numbers in (1) because they
are not square-free.

I do not know whether my above answer is accurate for your question,
but please let me know if I have misinterpreted your question.

With best wishes,

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

> It is also possible to check each of the statements in the paper for
> several randomly chosen variables to see if they are true.
> There must be an error somewhere or an implicit restriction.
> Max
>
> On Fri, Aug 3, 2012 at 5:21 PM, Paolo Lava <paoloplava at gmail.com> wrote:
> > 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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