A particular case of Dirichlet's theorem on arithmetic progressions
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(2) L'Enseignement Mathématique. Sedrakian, Naïri / Steinig, John. PARTICULAR CASE OF DIRICHLET'S THEOREM ON ARITHMETIC PROGRESSIONS. L'Enseignement Mathématique, Vol.44 (1998). PDF erstellt am: 28-gen-2010. Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch.
(3) A. PARTICULAR CASE OF. DIRICHLET'S THEOREM ON ARITHMETIC PROGRESSIONS by Naïri. that. p=a. a=. 1.. and John. Steinig. arithmetic progression states that if a relatively prime integers, there exist infinitely many primes p such (mod m) We give hère an elementary proof of the case in which. Dirichlet's theorem and m are. Sedrakian. on primes in an. .. following notation. If. positive integers, with ,Jt r ] (xi,...,x r ) dénotes their greatest common divisor and |>i, their least common multiple. For r — 1, we set (x\) := x\ and [x\] := x\ We. use. the. x\. ,. .. .. .. ,. xr. are. r>2,. .. .. .. .. The proof rests on three lemmas.. LEMMA 1. <. < r,. /. 1.. If. m. and a\,. .. .. .. ,a. r. are integers, with. m>l. and. a^. >1 for. then. (1). Proof. The case. Computing (m ai. r=l. - \,m ai -1). runs parallel to that of (a u the associative. (For. a. is. a. 2. trivial. The case r=2 can be established by with the euclidean algorithm; the computation ). One can then continue by induction, using. property. différent proof of the case. r. = 2, see [B], p. 26.).
(4) LEMMA. If x\.. 2.. .. .. .. ,x r are positive integers,. îhen. (2). where the numerator on the right hand side is îhe product of îhe gcd's of X\. ,x r taken nata îime for odd n— 1,3,... ; the denominator is the .. .. .. ,. the. product of There are. 22. r. ~l~. gcd's of x\. l. factors. .. .. the. r=l. The case. Proof. in. .. is. ,x r. taken. ,. nata. numerator and. trivial. For. 22 r. r=2,. time. for. n— 2, 4, the denominator.. even. —lin. ~1~ 1. identity. .. is. (2). .. .. .. familiar. the. (3). One can continue by induction, using (3) and the associative and distributive. properties. respectively. (Identity consists in. due. is. (2). to. V.-A.. Le. Besgue. ([3], pp. 51-53),. showing that any prime divides both sides of (2). whose. proof. the. same. to. power.) LEMMA. mbean. Let. 3.. integer,. m>l;. let p\.. .. .. .p r be distinct primes. .. which divide m. Then (4). it. m Proof Since m —lis divisible by each integer is divisible by their least common multiple. Hence. mm. m. (4). l pi. —I(i=. will. be. 1,. .. .. .. ,. r),. proved if we. can show that (5) is. impossible.. Xj. =. mm. m. lp. ' 1. this. To. —lin. occur. Since p\,. .. end,. Lemma .. .. -,p. r. are. we 2,. rewrite the left hand side of (5) by setting and then apply Lemma Ito the gcd's which. distinct primes, we hâve.
(5) This will bring (5) to the forai. (6). with. k=. 2. 1. '-. 1. and. -^^. m=. (j > 2).. <nj. But (6) would imply that. that is,. mni+lm. ni+1. m". |. 1. ;. this. is. since. impossible,. 1.. >. m. concludes the. This. proof. We can now prove the. Let m be an integer,. THEOREM. p. such that p =. Proof.. By. a. 1. m>. familiar argument [10],. =1 (mod p\m),. then. mbean. Now let. .. There exist infinitely many primes. (mod m).. each m > 1, of at least one prime p = /?2. 1. /?2. =1 (mod. integer,. divisors. Define the integer. TV. m>l,. it 1. m). suffices to prove the existence, for (mod m). (If pi = (mod m) and 1. and. Pi. /?2. and let p u. .. .. .. 7^. +1 >P\•). ,p s be its distinct prime. by. (7). Then. Af >. 1. by Lemma 3. Let. q. be any. prime divisor of N; we shall show. that (8). Since. q\N,. we hâve. (9) and. (10) It. follows from (10) that. (H). q. does not divide. m, whence.
(6) By (10), (11) and Lemma. 1,. (12). Suppose now that (8) does not hold. l. <i. Then. (m,. -. q. 1). |. —. for some. i,. Pi. < s, whence by (12),. (13) and therefore. (14). But (14) fact that. is q. impossible, for with (9) it implies that p = g, contradicting the does not divide m. This concludes the proof of the theorem. t. elementary proof s of this spécial case of Dirichlet's theorem are known; see [I], [2, §11.3], [4, §48], [s], [6], [7, §6.lA], [8, Ch. 6,5], [9], [10] and the références in [7, pp. 241-245]. They involve, more. Remark.. Several. Although the proof cyclotomic polynomials, say n (x). we hâve given hère does not require any knowledge of thèse polynomials, the integer TV defined in (7) is in fact equal to as can be seen with m (m),. or less. the. explicitly,. On(x).O. Om(m),O. Lemmas. where. \x. 1. is. and. and the identity [2, p. 181]. 2. the Môbius. function. (see also. §46). [4],. REFERENCES [1]. ESTERMANN,. T.. Note on. paper of A. Rotkiewicz. Acta Arithmetica. a. 8. (1963),. 465-467. [2] [3]. HASSE, H. Vorlesungen über Zahlentheorie.. Auflage. Springer-Verlag (Berlin, Gottingen, Heidelberg, New York), 1964. Le BESGUE, V.-A. Introduction àla théorie des nombres. Mallet-Bachelier 2.. (Paris), 1862. [4]. T.. NAGELL, Co.. [5]. Introduction. (New York),. to. Number Theory.. 2. nd. édition. Chelsea Publishing. 1964.. Primes in certain arithmetic progressions. Amer. Math. Monthly 83 (1976), 467-469.. NIVEN,. I.. and B.. POWELL..
(7) Démonstration arithmétique de l'existence d'une infinité de nombres premiers de la forme nk-\-l L'Enseignement Math. (2) 7 (1961), 277-280. SHAPIRO, H.N. Introduction to the Theory of Numbers. John Wiley & Sons, ROTKIEWICZ, A.. [6]. .. [7]. Inc. (New York),. 1983. nd. édition (éd. A. Schinzel). North-Holland (Amsterdam, New York, Oxford) and PWN (Warszawa),. SIERPINSKI, W. Elementary Theory of Numbers.. [8]. 2. 1988. SCHUR, I. Ùber die Existenz unendlich vieler. Primzahlen in einigen speziellen arithmetischen Progressionen. Sitzber. der Berliner Math. Ges. 11 (1912), 40-50. Reproduced in Gesammelte Abhandlungen II (éd. A. Brauer and H. Rohrbach), 1-11. Springer-Verlag (Berlin, Gottingen, New York),. [9]. 1973.. Wendt, E. Elementarer Beweis des Satzes, dass in jeder unbegrenzten arith unendlich viele Primzahlen vorkommen. metischenProgression my -f /. fur die reine und angewandte Math. 115 (1895), 85-88.. [10]. 1. (Reçu le 30 juin 1997;. Naïri Sedrakian c/o Vardan. Akopian. rue Francis de Crois set, A406 F-75018 Paris 8,. France e-mail : [email protected] John Steinig. Section. de. Université. mathématiques de. Genève. C.P 240. CH-1211 Genève 24 Switzerland. version révisée reçue. le. 30. avril 1998).
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