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Topic: Nico Benschop's Congruence.
Replies: 1   Last Post: Apr 6, 1999 6:00 PM

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Bill Dubuque

Posts: 156
Registered: 12/6/04
Re: Nico Benschop's Congruence.
Posted: Apr 6, 1999 6:00 PM
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jr@redmink.demon.co.uk (John R Ramsden) wrote:
|
| kramsay@aol.com (KRamsay) wrote:
| >
| > Nico Benschop supposed that for p prime, and 1<=a<b<=(p-1)/2,
| >
| > (a+1)^{p^(p-1)}-a^{p^(p-1)} <> (b+1)^{p^(p-1)}-b^{p^(p-1)} mod p^p,
| >
| > but so far as I know nobody's given a proof yet. I have a sketch [...]
|
| I have proved this with the help of results obtained by R D Carmichael
| using elementary but quite intricate arguments detailed in:
|
| "On the Numerical Factors of the Arithmetic Forms a^n +/- b^n",
| Annals of Math, Series 2, vol 15 (1913-14), pp 30 - 70.
|
| Among many other results, he showed that if a, b are coprime rational
| integers and n > 2 then a^n - b^n contains at least one "characteristic
| factor" except when n, a + b, a.b = 6, +/-3, 2.
|
| By a characteristic factor he meant a prime, p, which divides a^n - b^n
| but not a^m - b^m for any integer m dividing n. (Of course if m divides n
| then a^m - b^m divides a^n - b^n.)
|
| He also proved that _every_ prime dividing a^t - b^t, but not t,
| is a characteristic factor of a^s - b^s for some s dividing t.
| So in your problem we can assume that all prime factors other
| than p are characteristic.
|
| He further showed that a characteristic factor of a^t - b^t is of
| the form k.t + 1, so that the product of every such characteristic
| factor is also of this form. [...]

Note that the result is simply a statement about the existence
of a prime p such that, modulo p, a/b has order n (the modern
terminology is: p is a *primitive* prime factor of a^n - b^n).

This result is due to Zsigmondy (1892), with special cases (b=1)
(re)discovered by Bang (1886), Birkhoff & Vandiver (1904) ... see
Ribenboim, The New Book of Prime Number Records, p. 43,67-8,338,437.
Recall that Vandiver was a prolific researcher on FLT, e.g. see
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Vandiver.html
hence it is not surprising that these results were already applied
to Fermat's Last Theorem (first case) and related diophantine equations,
e.g. see Ribenboim's book 13 Lectures on FLT, pages 52,*161*,206,234,236.

Such results have many applications: a MathSciNet search on
Anywhere=(Zsigmondy or (Birkhoff and Vandiver)) will find
over 35 related Math Reviews. Schinzel (1974, MR 93k:11107)
extended the theorem to arbitrary algebraic number fields K:

If A and B are algebraic integers in K, (A,B)=1, A/B of degree d
and not a root of unity, then there exists n0 = n_0(d) such that
for all n > n0, A^n - B^n has a prime ideal factor P which
does not divide A^m - B^m for all m < n.

Later (1993, same MR) "he generalized this to show that every
algebraic number which is not a root of unity satisfies only a
finite number of independent generalized cyclotomic equations
considered by the reviewer [in Structural properties of
polylogarithms, Chapter 11, see p. 236, Amer. Math. Soc.,
Providence, RI, 1991; see MR 93b:11158]".

-Bill Dubuque






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