Theory Irregular_Primes_Infinite
section ‹Infinitude of irregular primes›
theory Irregular_Primes_Infinite
imports Regular_Primes
begin
text ‹
One consequence of Kummer's congruence is that there are infinitely many irregular primes.
We shall derive this here.
›
lemma zeta_real_gt_1:
assumes "x > 1"
shows "Re (zeta (of_real x)) > 1"
proof -
have *: "(λn. Re (complex_of_nat (Suc n) powr -of_real x)) sums (Re (zeta x))"
using assms by (intro sums_Re sums_zeta) auto
have **: "(Re (zeta x) - (1 + Re (2 powr -of_real x))) ≥ 0"
proof (rule sums_le)
show "(λn. Re (complex_of_nat (n+3) powr -of_real x)) sums (Re (zeta x) - (1 + Re (2 powr -of_real x)))"
using sums_split_initial_segment[OF *, of 2] by (simp add: eval_nat_numeral)
show "(λ_. 0) sums (0 :: real)"
by simp
next
fix n :: nat
have "complex_of_nat (n + 3) powr -complex_of_real x = of_real (real (n + 3) powr -x)"
by (subst powr_of_real [symmetric]) auto
also have "Re … = real (n + 3) powr -x"
by simp
also have "… ≥ 0"
by simp
finally show "Re (complex_of_nat (n + 3) powr -complex_of_real x) ≥ 0" .
qed
have "0 < Re (of_real (2 powr -x))"
by simp
also have "complex_of_real (2 powr -x) = 2 powr -complex_of_real x"
by (subst powr_of_real [symmetric]) auto
also have "Re (2 powr -of_real x) ≤ Re (zeta x) - 1"
using ** by simp
finally show ?thesis
by linarith
qed
lemma zeta_real_gt_1':
assumes "Re s > 1" "s ∈ ℝ"
shows "Re (zeta s) > 1"
using assms by (elim Reals_cases) (auto simp: zeta_real_gt_1)
lemma bernoulli_even_conv_zeta:
"complex_of_real (bernoulli (2*n)) = (-1)^Suc n * 2 * fact (2*n) / (2*pi)^(2*n) * zeta (2 * of_nat n)"
by (subst zeta_even_nat[of n]) (auto simp: field_simps)
lemma bernoulli_even_conv_zeta':
"bernoulli (2*n) = (-1)^Suc n * 2 * fact (2*n) / (2*pi)^(2*n) * Re (zeta (2 * of_nat n))"
proof -
have "complex_of_real (bernoulli (2*n)) = (-1)^Suc n * 2 * fact (2*n) / (2*pi)^(2*n) * zeta (2 * of_nat n)"
by (rule bernoulli_even_conv_zeta)
also have "… = of_real ((-1)^Suc n * 2 * fact (2*n) / (2*pi)^(2*n) * Re (zeta (2 * of_nat n)))"
using zeta_real'[of "2 * of_nat n"] by simp
finally show ?thesis
by (subst (asm) of_real_eq_iff)
qed
lemma abs_bernoulli_even_conv_zeta:
assumes "even n" "n > 0"
shows "¦bernoulli n¦ = 2 * fact n / (2*pi)^n * Re (zeta (of_nat n))"
proof -
from assms obtain k where n: "n = 2 * k"
by (elim evenE)
show ?thesis
using zeta_real_gt_1'[of n] assms(2) unfolding n bernoulli_even_conv_zeta'
by (simp add: abs_mult)
qed
lemma abs_bernoulli_over_n_ge_2:
assumes "n ≥ 23" "even n"
shows "¦bernoulli n / n¦ ≥ 2"
proof -
have "2 * real n ≤ 2 * real n * 1"
by simp
also have "… ≤ 2 * (fact n / (2*pi)^n) * Re (zeta n)"
using fact_ge_2pi_power[OF assms(1)] assms(1)
by (intro mult_left_mono mult_mono less_imp_le[OF zeta_real_gt_1']) (auto simp: field_simps)
also have "… = abs (bernoulli n)"
using assms by (subst abs_bernoulli_even_conv_zeta) auto
finally show ?thesis
using assms(1) by (simp add: field_simps)
qed
lemma infinite_irregular_primes_aux:
assumes "finite P" "∀p∈P. irregular_prime p" "37 ∈ P"
shows "∃p. irregular_prime p ∧ p ∉ P"
proof -
define n where "n = (∏p∈P. p - 1)"
have nz: "(∏p∈P. p - 1) ≠ 0"
using assms by (subst prod_zero_iff) (auto simp: irregular_prime_def)
define N where "N = bernoulli_num"
define D where "D = bernoulli_denom"
have "37 - 1 dvd (∏p∈P. p - 1)"
by (rule dvd_prodI) (use assms in auto)
hence "n ≥ 36"
unfolding n_def using nz by (intro dvd_imp_le) auto
have "even n"
unfolding n_def using assms by (auto simp: even_prod_iff intro!: bexI[of _ 37])
have [simp]: "N n ≠ 0"
using ‹even n› by (auto simp: N_def bernoulli_num_eq_0_iff)
have "abs (bernoulli n / n) > 1"
proof -
have "1 < (2 :: real)"
by simp
also have "… ≤ abs (bernoulli n / n)"
by (rule abs_bernoulli_over_n_ge_2) (use ‹even n› ‹n ≥ 36› in auto)
finally show ?thesis .
qed
obtain p where p: "prime p" "qmultiplicity (int p) (bernoulli n / of_nat n) > 0"
proof -
obtain a b where ab: "quotient_of (bernoulli n / of_nat n) = (a, b)"
using prod.exhaust by blast
have "b > 0"
using ab quotient_of_denom_pos by blast
have eq': "of_int a / of_int b = (bernoulli n / of_nat n :: rat)"
using ab quotient_of_div by simp
also have "(of_rat … :: real) = bernoulli n / n"
by (simp add: of_rat_divide of_rat_bernoulli)
finally have eq: "real_of_rat (rat_of_int a / rat_of_int b) = bernoulli n / real n" .
have [simp]: "a ≠ 0"
using eq ‹abs (bernoulli n / n) > 1› by auto
moreover have "¬is_unit a "
proof
assume "is_unit a"
hence "abs (bernoulli n / n) = 1 / of_int b"
unfolding eq [symmetric] using ‹b > 0› by (simp add: of_rat_divide flip: of_int_abs)
also have "… ≤ 1"
using ‹b > 0› by simp
finally show False
using ‹abs (bernoulli n / n) > 1› by simp
qed
ultimately obtain p' where p': "prime p'" "p' dvd a"
using prime_divisor_exists by blast
define p where "p = nat p'"
from p' have p: "prime p" "int p dvd a"
unfolding p_def by (auto intro: prime_ge_0_int)
show ?thesis
proof (rule that[of p])
from p have "multiplicity p a ≥ 1"
by (intro multiplicity_geI) auto
moreover have "coprime a b"
using ab quotient_of_coprime by auto
hence "¬p dvd b"
using ab p by (meson not_coprimeI not_prime_unit prime_nat_int_transfer)
hence "multiplicity p b = 0"
by (intro not_dvd_imp_multiplicity_0) auto
ultimately show "qmultiplicity (int p) (bernoulli n / of_nat n) > 0"
using ‹b > 0› p unfolding eq'[symmetric] by (subst qmultiplicity_divide_of_int) auto
qed fact+
qed
hence "qmultiplicity p (of_int (N n) / of_int (int (D n * n))) > 0"
by (simp add: N_def D_def bernoulli_conv_num_denom)
hence "multiplicity (int p) (N n) > 0"
unfolding bernoulli_conv_num_denom using p
by (subst (asm) qmultiplicity_divide_of_int)
(use ‹n ≥ 36› ‹even n› in ‹simp_all add: N_def D_def bernoulli_num_eq_0_iff›)
hence "p dvd N n"
using not_dvd_imp_multiplicity_0 by fastforce
have "¬(p - 1) dvd n"
proof
assume "(p - 1) dvd n"
hence "p dvd D n"
using p(1) ‹even n› ‹n ≥ 36› unfolding D_def by (subst prime_dvd_bernoulli_denom_iff) auto
hence "¬p dvd N n" unfolding N_def D_def
using coprime_bernoulli_num_denom[of n] p(1)
by (metis coprime_def int_dvd_int_iff not_prime_unit of_nat_1)
with ‹p dvd N n› show False
by contradiction
qed
hence "p ∉ P" "p ≠ 2"
unfolding n_def using assms by auto
hence "p > 2"
using prime_gt_1_nat[OF ‹prime p›] by linarith
hence "odd p"
using prime_odd_nat[of p] ‹prime p› by auto
define r where "r = n mod (p - 1)"
have "even (n - n div (p - 1) * (p - 1))"
using ‹p > 2› ‹odd p› ‹even n› by (subst even_diff_nat) auto
also have "… = r"
unfolding r_def by (rule minus_div_mult_eq_mod)
finally have "even r" .
have "irregular_prime p"
proof (rule irregular_primeI)
have "r > 0" "r < p - 1"
using p prime_gt_1_nat[of p] ‹¬(p - 1) dvd n›
unfolding r_def by (auto intro!: pos_mod_bound Nat.gr0I)
moreover from ‹even r› and ‹prime p› have "r ≠ 1" "r ≠ p - 2"
using ‹odd p› ‹p > 2› by auto
ultimately have "r ≥ 2" "r ≤ p - 3"
by linarith+
thus "r ∈ {2..p-3}"
by auto
have "bernoulli r / of_nat r = (0 :: rat) ∨
qmultiplicity p (bernoulli r / of_nat r) > 0"
proof (rule qcong_qmultiplicity_pos_transfer)
show "qmultiplicity p (bernoulli n / of_nat n) > 0"
using p by simp
next
have "[n = r] (mod (p - 1))"
by (auto simp: Cong.cong_def r_def)
thus "[bernoulli n / of_nat n = bernoulli r / of_nat r] (qmod p)"
using p(1) ‹even n› ‹even r› ‹n ≥ _› ‹r > 0› ‹¬(p-1) dvd n› unfolding D_def N_def
by (intro kummer_congruence'_prime) auto
qed
moreover have "bernoulli r / of_nat r ≠ (0 :: rat)"
using ‹even r› ‹r > 0› by (auto simp: bernoulli_zero_iff)
ultimately have "qmultiplicity p (bernoulli r / of_nat r) > 0"
by simp
hence "qmultiplicity p (of_int (N r) / of_int (int (D r * r))) > 0"
by (simp add: bernoulli_conv_num_denom N_def D_def)
hence "multiplicity (int p) (N r) > 0"
by (subst (asm) qmultiplicity_divide_of_int)
(use ‹r > 0› ‹even r› ‹prime p› in ‹simp_all add: N_def D_def bernoulli_num_eq_0_iff›)
hence "p dvd N r"
using not_dvd_imp_multiplicity_0 by fastforce
thus "p dvd bernoulli_num r"
by (simp add: N_def)
qed (use p ‹p ≠ 2› ‹even r› in auto)
with ‹p ∉ P› show ?thesis
by blast
qed
theorem infinite_irregular_primes: "infinite {p. irregular_prime p}"
proof
assume "finite {p. irregular_prime p}"
hence "∃p. irregular_prime p ∧ p ∉ {p. irregular_prime p}"
by (rule infinite_irregular_primes_aux) (use irregular_prime_37 in auto)
thus False
by simp
qed
end