http://vamri.xray.ufl.edu/proths/status.html
where I found:
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Proth's Theorem (1878): Let n = k.(2^q)+1 with 2^q > k. If there is an
integer $a$ such that a^((n-1)/2) = -1 (mod n), then $n$ is prime.
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It may be interesting to notice that, in his "The/orie des nombres"
(Paris, Gauthier-Villars, 1891), p. 441, Edouard Lucas claims he stated
for the first time in 1876 the following (famous) theorem:
*If a^x = 1 (mod n) for x = n-1 and for no other divisor of (n-1), then
the number $n$ is prime.*
of which Proth's is a quite obvious consequence.
In the following, I give the original text by Lucas (in French):
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(...) on a donc la proposition suivante, que l'on doit conside/rer comme
la re/ciproque du the/ore\me de Fermat: Si a^x - 1 est divisible par $n$,
pour $x$ e/gal a\ (n-1), et n'est pas divisible par $n$, pour $x$ e/gal
a\ une partie aliquote de (n-1), le nombre $n$ est premier.
Nous avons e/nonce/ pour la premie\re fois ce the/ore\me en 1876, au
Congre\s de l'Association franc,aise pour l'Avancement des Sciences, a\
Clermont-Ferrand, dans une Note intitule/e: Sur la recherche des grands
nombres premiers.
*************
Does anyone know about this question of priority? By the way, who Proth
was, and where did he publish his (?) theorem?
Udai Venedem
venedem@wanadoo.fr
http://perso.wanadoo.fr/alta.mathematica/