Primality tests for specific classes ofn = k 2 m ±1

Transcription

1 Primality tests for specific classes ofn k 2 m ±1 Predrag Terzić Bulevar Pera Ćetkovića 19, Podgorica, Moteegro pedja.terzic@hotmail.com arxiv: v1 [math.nt] 10 Ju 2015 Abstract: Polyomial time primality tests for specific classes of umbers of the form k 2 m ±1 are itroduced. Keywords: Primality test, Polyomial time, Prime umbers. AMS Classificatio: 11A51. 1 Itroductio I 1856 Edouard Lucas developed primality test for Mersee umbers. The test was improved by Lucas i 1878 ad Derrick Hery Lehmer i the 190s, see [1].I 1960 Kusta Ikeri provided ucoditioal, determiistic, lucasia type primality test for Fermat umbers, see [2].I 1969 Has Riesel formulated primality test, see [] for umbers of the formk 2 1 withk odd ad k < 2. I 2006 Zhi-Hog Su provided geeral primality criterio for umbers of the form k 2 m ± 1,see [4]. I this ote I preset lucasia type primality tests for specific classes of k 2 m ±1. Throughout this paper we use the followig otatios: Z-the set of itegers, N-the set of positive itegers, a p -the Jacobi symbol, m,-the greatest commo divisor of m ad, S x-the sequece defied by S 0 x x ad S k+1 x S k x 2 2k 0. 2 Basic Lemmas ad Theorems Defiitio 2.1. ForP,Q Z the Lucas sequece{v P,Q} is defied by V 0 P,Q 2,V 1 P,Q P,V +1 P,Q PV P,Q QV 1 P,Q 1 Let D P 2 4Q. It is kow that V P,Q Lemma 2.1. Let P,Q Z ad N. The V P,Q [/2] r0 P+ D 2 + r r r P D 2 P 2r Q r 1

2 Theorem 2.1. Zhi-Hog Su For m {2,,4,...} let p k 2 m ± 1 with 0 < k < 2 m ad k odd. If b,c Z, 2c+b 2c b c p,c 1 ad the p is prime if ad oly if p S p p p m 2 x, where k 1/2 x c k V k b,c 2 k k r 1 r b/c k 2r k r r r0 For proof see Theorem.1 i [4]. Lemma 2.2. Let be odd positive umber, the 1 1, if 1 mod 4 1, if mod 4 Lemma 2.. Let be odd positive umber, the 2 1, if 1,7 mod 8 1, if,5 mod 8 Lemma 2.4. Let be odd positive umber, the case 1. 1 mod 4 case 2. mod 4 1 if 1 mod 0 if 0 mod 1 if 2 mod 1 if 2 mod 0 if 0 mod 1 if 1 mod Proof. Sice mod 4 if we apply the law of quadratic reciprocity we have two cases. If 1 mod 4 the ad the result follows. If mod 4 the ad the result follows. Lemma 2.5. Let be odd positive umber, the 5 1 if 1,4 mod 5 0 if 0 mod 5 1 if 2, mod 5 Proof. Sice 5 1 mod 4 if we apply the law of quadratic reciprocity we have 5 5 ad the result follows. Lemma 2.6. Let be odd positive umber, the case 1. 1 mod 4 2

3 1 if 1,11 mod 12 0 if,9 mod 12 1 if 5,7 mod 12 case 2. mod 4 Proof. 1 if 5,7 mod 12 0 if,9 mod 12 1 if 1,11 mod Applyig the law of quadratic reciprocity we have : if 1 mod 4 the. If mod 4 the. Applyig the Chiese remaider theorem i both cases several times we get the result. Lemma 2.7. Let be odd positive umber, the case 1. 1 mod 4 case 2. mod if 1,2,4 mod 7 0 if 0 mod 7 1 if,5,6 mod 7 1 if,5,6 mod 7 0 if 0 mod 7 1 if 1,2,4 mod 7 Proof. Sice 7 mod 4 if we apply the law of quadratic reciprocity we have two cases. If 1 mod 4 the 7 7 ad the result follows. If mod 4 the 7 7 ad the result follows. Lemma 2.8. Let be odd positive umber, the case 1. 1 mod 4 case 2. mod if 1,5,19,2 mod 24 0 if,9,15,21 mod 24 1 if 7,11,1,17 mod if 7,11,1,17 mod 24 0 if,9,15,21 mod 24 1 if 1,5,19,2 mod 24

4 Proof Applyig the law of quadratic reciprocity we have : if 1 mod 4 the. If mod 4 the. Applyig the Chiese remaider theorem i both cases several times we get the result. Lemma 2.9. Let be odd positive umber, the 10 1 if 1,,9,1,27,1,7,9 mod 40 0 if 5,15,25,5 mod 40 1 if 7,11,17,19,21,2,29, mod 40 Proof Applyig the law of quadratic reciprocity we have : 5 5. Applyig the Chiese remaider theorem several times we get the result. The Mai Result Theorem.1. Let N k 2 m 1such thatm > 2, k, 0 < k < 2 m ad k 1 mod 10 with m 2, mod 4 k mod 10 with m 0, mod 4 k 7 mod 10 with m 1,2 mod 4 k 9 mod 10 with m 0,1 mod 4 Let b ad S 0 x V k b,1, thus N is prime iff N S m 2 x Proof. Sice N mod 4 ad b from Lemma 2.2 we kow that N 1. Similarly, sice N 2 mod 5 or N mod 5 ad b from Lemma 2.5 we kow that 2+b N 1. From Lemma 2.1 we kow that Vk b,1 x. Applyig Theorem 2.1 i the case c 1 we get the result. Theorem.2. Let N k 2 m 1such thatm > 2, k, 0 < k < 2 m ad k mod 42with m 0,2 mod k 9 mod 42with m 0 mod k 15 mod 42with m 1 mod k 27 mod 42with m 1,2 mod k mod 42with m 0,1 mod k 9 mod 42with m 2 mod Let b 5 ad S 0 x V k b,1, thus N is prime iff N S m 2 x 4

5 Proof. Sice N mod 4 ad N 11 mod 12 ad b 5 from Lemma 2.6 we kow that N 1. Similarly, sicen mod 4 ad N 1 mod 7 orn 2 mod 7 or N 4 mod 7 ad b 5 from Lemma 2.7 we kow that 2+b N 1. From Lemma 2.1 we kow thatv k b,1 x. Applyig Theorem 2.1 i the case c 1 we get the result. Theorem.. Let N k 2 m +1 such thatm > 2, 0 < k < 2 m ad k 1 mod 42with m 2,4 mod 6 k 5 mod 42with m mod 6 k 11 mod 42with m,5 mod 6 k 1 mod 42with m 4 mod 6 k 17 mod 42with m 5 mod 6 k 19 mod 42with m 0 mod 6 k 2 mod 42with m 1, mod 6 k 25 mod 42with m 0,2 mod 6 k 29 mod 42with m 1,5 mod 6 k 1 mod 42with m 2 mod 6 k 7 mod 42with m 0,4 mod 6 k 41 mod 42with m 1 mod 6 Let b 5 ad S 0 x V k b,1, thus N is prime iff N S m 2 x Proof. Sice N 1 mod 4 ad N 5 mod 12 ad b 5 from Lemma 2.6 we kow that N 1. Similarly, sicen 1 mod 4 ad N mod 7 orn 5 mod 7 or N 6 mod 7 ad b 5 from Lemma 2.7 we kow that 2+b N 1. From Lemma 2.1 we kow thatv k b,1 x. Applyig Theorem 2.1 i the case c 1 we get the result. Theorem.4. Let N k 2 m +1 such thatm > 2, 0 < k < 2 m ad k 1 mod 6 ad k 1,7 mod 10 withm 0 mod 4 k 5 mod 6 ad k 1, mod 10 withm 1 mod 4 k 1 mod 6 ad k,9 mod 10 withm 2 mod 4 k 5 mod 6 ad k 7,9 mod 10 withm mod 4 Let b 8 ad S 0 x V k b,1, thus N is prime iff N S m 2 x Proof. Sice N 1 mod 4 ad N 17 mod 24 ad b 8 from Lemma 2.8 we kow that N 1. Similarly, sice N 17 mod 40 or N mod 40 ad b 8 from Lemma 2.9 we kow that 2+b N 1. From Lemma 2.1 we kow that Vk b,1 x. Applyig Theorem 2.1 i the casec 1 we get the result. 5

6 Refereces [1] Cradall, Richard; Pomerace, Carl 2001, Sectio 4.2.1: The Lucas-Lehmer test, Prime Numbers: A Computatioal Perspective 1st ed., Berli: Spriger, p [2] Ikeri, K., Tests for primality, A. Acad. Sci. Fe., A I 279, [] Riesel, Has 1969, Lucasia Criteria for the Primality ofn h 2 1, Mathematics of Computatio AmericaMathematical Society, 2 108: [4] Zhi-Hog Su, Primality tests for umbers of the formk 2 m ±1, The Fiboacci Quarterly, , o.2,