A LLT-like test for proving the primality of Fermat numbers.

Transcription

1 A LLT-like test for roving the rimality of Fermat numbers. Tony Reix First version: 004, 4th of Setember Udated: 005, 9th of October Revised (Inkeri): 009, 8th of December In 876, Édouard Lucas discovered a method for roving that a number is rime or comosite without searching its factors. His method was based on the roerties of the Lucas Sequences. He first used his method for Mersenne numbers and roved that 7 is a rime. In 930, Derrick Lehmer rovided a comlete and clean roof. This test of rimality for Mersenne numbers is now known as: Lucas-Lehmer Test (LLT). Few eole know that Lucas also used his method for roving that a Fermat number is rime or comosite, still with an unclear roof. He used his method for roving that 6 + is comosite. Lehmer did not rovide a roof of Lucas method for Fermat numbers. This aer rovides a roof of a LLT-like test for Fermat numbers, based on the roerties of Lucas Sequences and based on the method of Lehmer. The seed (the starting value S 0 of the {S i } sequence) used here is 5, though Lucas used 6. In 960, Kustaa Inkeri rovided a full roof with seed 8. Primality tests for secial numbers are classified into N and N + categories, meaning that the numbers N or N + can be comletely or artially factored. Since many books talk about the LLT only in the N + chater for Mersenne numbers N = q, it seemed useful to remind that the LLT can also be used for numbers N such that N is easy to factor, like Fermat numbers N = n +, by roviding a roof àla Lehmer. Theorem F n = n + (n ) is a rime if and only if it divides S n, where S 0 = 5 and S i = S i for i =,, 3,... n. The roof is based on chaters 4 (The Lucas Functions) and 8.4 (The Lehmer Functions) of the book Édouard Lucas and Primality Testing of H. C. Williams (A Wiley-Interscience ublication, 998). Chater exlains how the (P, Q) arameters have been found. Then Chater rovides the Lehmer theorems used for the roof. Then Chater 3 and 4 rovide the roof for: F n rime = F n S n and the converse, roving theorem. Chater 5 rovides numerical examles. The aendix in Chater 6 rovides first values of U n and V n lus some roerties. AMS Classification: A5 (Primality), B39 (Lucas Sequences), -03 (Historical), 0A55 (9th century), 0A60 (0th century).

2 Lucas Sequence with P = R Let S 0 = 5 and S i = S i. S = 3, S = 57 = 7 3,... It has been checked that: { S n 0 (mod F n ) for n =...4 S n 0 (mod F n ) for n = Here after, we search a Lucas Sequence (U m ) m 0 and its comanion (V m ) m 0 with (P, Q) that fit with the values of the S i sequence. We define the Lucas Sequence V m such that: Thus we have: V k+ = S k () V = S 0 = 5 V 4 = S = 3 V 8 = S = 57 If (4..7) age 74 ( V n = V n Q n ) alies, we have: and thus: Q = V V 4 = 4 V4 V 8 = ±. { V4 = V Q V 8 = V 4 Q4 With (4..3) age 70 ( V n+ = P V n QV n ), and with: V 0 = V = P V = P V QV 0 = P Q we have: P = V + Q = 7 or 3. In the following we consider: (P, Q) = ( 7, ). As exlained by Williams age 96, all of the identity relations [Lucas functions] given in (4.) continue to hold, as these are true quite without regard as to whether P, Q are integers. So, like Lehmer, we define P = R such that R = 7 and Q = are corime integers and we define (Proerty (8.4.) age 96): { Vn when n { Un / R when n V n = V n / R when n U n = U n when n in such a way that V n and U n are always integers. Tables to 5 give values of U i, V i, U i (mod F n ), V i (mod F n ), with (P, Q) = ( 7, ), for n =,, 3, 4.

3 Lehmer theorems Like Lehmer, let define the symbols (where ( a/b) is the Legendre symbol): ε = ε() = ( ) D/ σ = σ() = ( ) R/ τ = τ() = ( ) Q/ The following formulas (from age 77) will hel roving roerties: (4..8) m U mn = (4..9) m V mn = Proerty (8.4.) age 96 : m m If is an odd rime and Q, then: ( ) m i + ( ) m D i Un i Vn m i i D i Un i+ Vn m (i+) { U ( ) D/ (mod ) V ( R/) (mod ) Proof: Since is a rime, and by Fermat little theorem, we have: (mod ). By (4..8), with m = and n =, since U = and V = P, we have: U = U = P + D i U i+ V (i+) i + DP D P 0 Since ( ( i) 0 (mod ) when 0 < i < and ) =, we have: U = U D ( ) D/ (mod ) By (4..9), with m = and n =, since U = and V = P, we have: V = V = P + 0 ( D i U i V i i ) DP D P Since ( ( 0) =, and i) 0 (mod ) when 0 < i <, we have: V P and V P R ( ) R/ (mod ) 3

4 Proerty (8.4.3) age 97 : odd rime and Q = U σε Proof By (4..8) with n =, V = P, since is a rime and (R, Q) =, we have: With: m = + U + = + ( ) ( ) + + U + = P + DP D i P i i + ( + ) ( D + P + + U + = ( + )P + ( + ) [... ] + ( + )D + P + 0D P U + = P + D [ ] [ P +... = P (P ) + D ] [ ] +... ) D + P U + P = U + R + D ( R/) + ( D /) = σ() + ε() (mod ) Thus, if σε = σ() ε() =, then U + = U σɛ. With: m = : U = D i P (i+) i + ( ) ( ) ( ) U = P + DP D 3 P + D P U = ( )P + ( )DP ( )D 3 P + 0D P U P [P 3 + DP D 3 ] P D P D U (P D) (P ) + D ε() σ() (mod ) (mod ) Thus, if σε = σ() ε() =, then U = U σɛ. Proerty (8.4.4) age 97 If is an odd rime and Q, then: V σε σq σε (mod ). 4

5 Theorem (8.4.) If is an odd rime and QRD, then: V σɛ when σ = τ U σɛ when σ = τ Definition (8.4.) age 97 of ω(m) : For a given m, denote by ω = ω(m) the value of the least ositive integer k such that m U k. If ω(m) exists, ω(m) is called the rank of aarition of m. Theorem 3 (8.4.3) { If k n, then U k U n. If m U n, then ω(m) n. Theorem 4 (8.4.5) If (m, Q) =, then ω(m) exists. Theorem 5 (8.4.6) If (N, QRD) = and N ± is the rank of aarition of N, then N is a rime. Theorem 6 (8.4.7) If (N, QRD) =, U N± 0 (mod N) and U N± 0 (mod N) for each distinct rime divisor q of N ±, then N q is a rime. Proof: Let ω = ω(n). We see that ω N ±, but ω (N ± )/q. Thus if q α N ±, then q α ω. It follows that ω = N ± and N is a rime by Theorem 5 (8.4.6). 3 F n rime = F n V F n and F n S n Let N = F n = n + with n be an odd rime. Let: P = R, R = 7, Q =, and D = P 4Q = 3. Hereafter we comute ( ( 3/N) and 7 /N) : ( 3/N) : N odd rime N = (4) n + (mod 3) Since: ( ) ( ) N /3 = /3 = ( ) ( 3 3 /N = N /3) ( ) N then: ( 3/N) =. 5

6 ( 7/N) : We have: { 3 (mod 7) 3a+b b (mod 7) With n b (mod 3), we have: n + b + (mod 7). Then we study the exonents of, modulo 3. We have: (mod 3), and: m (mod 3) If n = m N = m (mod 7) ( ) ( ) N /7 = 3 /7 = If n = m + m+ (mod 3) N = m (mod 7) ( ) ( ) N /7 = 5 /7 = Finally, we have: ( ( ) 7 7/N) = N /7 ( ) n = ( N/7) =. So we have: ε = ( D/N) = ( 3 /N) = σ = ( R/N) = ( 7 /N) = τ = ( Q/N) = ( /N) = + Since σ = τ, σɛ = +, and F n QRD with n, then by Theorem (8.4.) we have: F n rime = F n V Fn = V n By () we have: V k = S k and thus, with k = n : F n S n. 4 F n S n = F n is a rime Let N = F n with n. By () we have: N S n = N V n. And thus, by (4..6) age 74 ( U a = U a V a ), we have: N U n. By (4.3.6) age 85: ( (V n, U n ) Q n for any n ), and since Q =, then: (V n, U n ) = and thus: N U n since N odd. With ω = ω(n), by Theorem 3 (8.4.3) we have : ω n and ω n. This imlies: ω = n = N. Then N is the rank of aarition of N, and thus by Theorem 5 (8.4.6) N is a rime. 6

7 This test of rimality for Fermat numbers has been communicated to the community of number theorists working on this area on mersenneforum.org (htt:// in May 004, and the roof was finalized in Setember 004. Then, in a rivate communication, Robert Gerbicz rovided a roof of the same theorem based on Q[ ]. 5 Numerical Examles (mod F ) S 0 = 5 6 S 0 (mod F 3 ) S 0 = = 60 6 S 3 0 (mod F 4 ) S 0 = S Aendix: Table of U i and V i With n =, 3, 4, we have the following (not roven) roerties (modulo F n ): U F n 5 5 V F n 5 3 U Fn 4 6 U F n 3 U Fn U F n 0 U F n U Fn+ U F n+ 6 U F n+3 5 V Fn 4 4 V F n 3 5 V Fn V F n V F n V Fn+ 5 V F n+ 4 V F n+3 3 The values of U n and V n ( n ) with (P, Q) = ( 3, ) can be built by: { U n = U n { V n = V n U n+ = V n+ V n+ = U n+ Values of U i and V i in revious tables can be comuted easily by the following PARI/g rograms: U j+ : U0=;U=6; for(i=,n, U0=5*U-U0; U=5*U0-U; rint(4*i+," ",U0); rint(4*i+," ",U)) 7

8 i U i V i Table : P = 7, Q = 8

9 i U i (mod F ) V i (mod F ) Table : P = 7, Q =, Modulo F i U i (mod F ) V i (mod F ) Table 3: P = 7, Q =, Modulo F 9

10 i U i (mod F 3 ) V i (mod F 3 ) Table 4: P = 7, Q =, Modulo F 3 0

11 i U i (mod F 4 ) V i (mod F 4 ) Table 5: P = 7, Q =, Modulo F 4