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Diffstat (limited to 'coqprime/PrimalityTest/Pepin.v')
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diff --git a/coqprime/PrimalityTest/Pepin.v b/coqprime/PrimalityTest/Pepin.v new file mode 100644 index 000000000..c400e0a43 --- /dev/null +++ b/coqprime/PrimalityTest/Pepin.v @@ -0,0 +1,123 @@ + +(*************************************************************) +(* This file is distributed under the terms of the *) +(* GNU Lesser General Public License Version 2.1 *) +(*************************************************************) +(* Benjamin.Gregoire@inria.fr Laurent.Thery@inria.fr *) +(*************************************************************) + +(********************************************************************** + Pepin.v + + Pepin's Test for Fermat Number + + Definition: PepinTest + **********************************************************************) +Require Import ZArith. +Require Import ZCAux. +Require Import Pocklington. + +Open Scope Z_scope. + +Definition FermatNumber n := 2^(2^(Z_of_nat n)) + 1. + +Theorem Fermat_pos: forall n, 1 < FermatNumber n. +unfold FermatNumber; intros n; apply Zle_lt_trans with (2 ^ 2 ^(Z_of_nat n)); auto with zarith. +rewrite <- (Zpower_0_r 2); auto with zarith. +apply Zpower_le_monotone; try split; auto with zarith. +Qed. + +Theorem PepinTest: forall n, let Fn := FermatNumber n in (3 ^ ((Fn - 1) / 2) + 1) mod Fn = 0 -> prime Fn. +intros n Fn H. +assert (Hn: 1 < Fn). +unfold Fn; apply Fermat_pos. +apply PocklingtonCorollary1 with (F1 := 2^(2^(Z_of_nat n))) (R1 := 1); auto with zarith. +2: unfold Fn, FermatNumber; auto with zarith. +apply Zlt_le_trans with (2 ^ 1); auto with zarith. +rewrite Zpower_1_r; auto with zarith. +apply Zpower_le_monotone; try split; auto with zarith. +rewrite <- (Zpower_0_r 2); apply Zpower_le_monotone; try split; auto with zarith. +unfold Fn, FermatNumber. +assert (H1: 2 <= 2 ^ 2 ^ Z_of_nat n). +pattern 2 at 1; rewrite <- (Zpower_1_r 2); auto with zarith. +apply Zpower_le_monotone; split; auto with zarith. +rewrite <- (Zpower_0_r 2); apply Zpower_le_monotone; try split; auto with zarith. +apply Zlt_le_trans with (2 * 2 ^2 ^Z_of_nat n). +assert (tmp: forall p, 2 * p = p + p); auto with zarith. +apply Zmult_le_compat_r; auto with zarith. +assert (Hd: (2 | Fn - 1)). +exists (2 ^ (2^(Z_of_nat n) - 1)). +pattern 2 at 3; rewrite <- (Zpower_1_r 2). +rewrite <- Zpower_exp; auto with zarith. +assert (tmp: forall p, p = (p - 1) +1); auto with zarith; rewrite <- tmp. +unfold Fn, FermatNumber; ring. +assert (0 < 2 ^ Z_of_nat n); auto with zarith. +intros p Hp Hp1; exists 3; split; auto with zarith; split; auto. +rewrite (Zdivide_Zdiv_eq 2 (Fn -1)); auto with zarith. +rewrite Zmult_comm; rewrite Zpower_mult; auto with zarith. +rewrite Zpower_mod; auto with zarith. +assert (tmp: forall p, p = (p + 1) -1); auto with zarith; rewrite (fun x => (tmp (3 ^ x))). +rewrite Zminus_mod; auto with zarith. +rewrite H. +rewrite (Zmod_small 1); auto with zarith. +rewrite <- Zpower_mod; auto with zarith. +rewrite Zmod_small; auto with zarith. +simpl; unfold Zpower_pos; simpl; auto with zarith. +apply Z_div_pos; auto with zarith. +apply Zis_gcd_gcd; auto with zarith. +apply Zis_gcd_intro; auto with zarith. +intros x HD1 HD2. +assert (Hd1: p = 2). +apply prime_div_Zpower_prime with (4 := Hp1); auto with zarith. +apply prime_2. +assert (Hd2: (x | 2)). +replace 2 with ((3 ^ ((Fn - 1) / 2) + 1) - (3 ^ ((Fn - 1) / 2) - 1)); auto with zarith. +apply Zdivide_minus_l; auto. +apply Zdivide_trans with (1 := HD2). +apply Zmod_divide; auto with zarith. +rewrite <- Hd1; auto. +replace 1 with (Fn - (Fn - 1)); auto with zarith. +apply Zdivide_minus_l; auto. +apply Zdivide_trans with (1 := Hd2); auto. +Qed. + +(* An optimized version with Zpow_mod *) + +Definition pepin_test n := + let Fn := FermatNumber n in if Z_eq_dec (Zpow_mod 3 ((Fn - 1) / 2) Fn) (Fn - 1) then true else false. + +Theorem PepinTestOp: forall n, pepin_test n = true -> prime (FermatNumber n). +intros n; unfold pepin_test. +match goal with |- context[if ?X then _ else _] => case X end; try (intros; discriminate). +intros H1 _; apply PepinTest. +generalize (Fermat_pos n); intros H2. +rewrite Zplus_mod; auto with zarith. +rewrite <- Zpow_mod_Zpower_correct; auto with zarith. +rewrite H1. +rewrite (Zmod_small 1); auto with zarith. +replace (FermatNumber n - 1 + 1) with (FermatNumber n); auto with zarith. +apply Zdivide_mod; auto with zarith. +apply Z_div_pos; auto with zarith. +Qed. + +Theorem prime5: prime 5. +exact (PepinTestOp 1 (refl_equal _)). +Qed. + +Theorem prime17: prime 17. +exact (PepinTestOp 2 (refl_equal _)). +Qed. + +Theorem prime257: prime 257. +exact (PepinTestOp 3 (refl_equal _)). +Qed. + +Theorem prime65537: prime 65537. +exact (PepinTestOp 4 (refl_equal _)). +Qed. + +(* Too tough !! +Theorem prime4294967297: prime 4294967297. +refine (PepinTestOp 5 (refl_equal _)). +Qed. +*) |