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Re: Nico Benschop's Congruence.
Posted:
Apr 6, 1999 6:00 PM


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<=(p1)/2,  >  > (a+1)^{p^(p1)}a^{p^(p1)} <> (b+1)^{p^(p1)}b^{p^(p1)} 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 (191314), 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. 43,678,338,437. Recall that Vandiver was a prolific researcher on FLT, e.g. see http://wwwgroups.dcs.stand.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



