[HM] Proth and Lucas
Udai Venedem (venedem@wanadoo.fr)
Tue, 10 Aug 1999 22:47:51 +0200
Dear friends,
I have been probably wrong interpreting Proth's as an "obvious consequence"
of Lucas' Theorem, and Avinoam Mann was quite right to be doubtful about it.
Nevertheless it was not useless to inquire about that matter, because it
allowed to bring more precision.
In particular, and this precision counts for Alfred Ross: if Comptes Rendus
Acad. Sci. Paris, 1878, t. LXXXVII, Seance du 2 Septembre 1878, p. 374,
states a theorem by "M. F. Proth", it is another one, much like Pepin's test.
The Proth's theorem we speak about is found in Comptes Rendus Acad. Sci.
Paris, 1878, t. LXXXVII, Seance du 9 Decembre 1878, p. 926, and it is one
of four theorems. Unhappily, here like in the former, he does not give any
demonstration.
But there were bel et bien a discussion between Proth and Lucas. It came
about various statements Proth announced on December 27th, 1876 (C. R., t.
LXXXIII), among which:
-If P is prime, the number (2^P - 2) is divisible by P; but not by P^2, nor
by P^3.
-The number (a^k - 1) may be prime only if a = 2 and k prime. The number
(a^k + 1) may be prime only if a = 2 and k = 2^n.
And Ed. Lucas, on January 8th, 1877 protests: "(il) adresse quelques
observations critiques, au sujet des e/nonce/s de the/ore\me sur les nombres
qui ont e/te/ communique/s par M. F. Proth, dans la se/ance du 27 de/cembre
1876." Without saying more.
But we cannot avoid thinking to this incident when we read in Lucas' The/orie
des nombres (Paris, Gauthier-Villars, 1891) p. 423:
"Existe-t-il un nombre premier p, tel que (2^(p-1) - 1) soit divisible par
p^2 ?"
And one knows, since Meissner (1913), this happens for p = 1093.
Good Heavens !
Udai Venedem
venedem@wanadoo.fr
http://perso.wanadoo.fr/alta.mathematica/