Theory Prime_Distribution_Elementary.Lcm_Nat_Upto
section ‹The LCM of the first $n$ natural numbers›
theory Lcm_Nat_Upto
imports Prime_Number_Theorem.Prime_Counting_Functions
begin
text ‹
In this section, we examine @{term "Lcm {1..(n::nat)}"}. In particular,
we will show that it is equal to $e^{\psi(n)}$ and thus (by the PNT) $e^{n + o(n)}$.
›
lemma multiplicity_Lcm:
fixes A :: "'a :: {semiring_Gcd, factorial_semiring_gcd} set"
assumes "finite A" "A ≠ {}" "prime p" "0 ∉ A"
shows "multiplicity p (Lcm A) = Max (multiplicity p ` A)"
using assms
proof (induction A rule: finite_ne_induct)
case (insert x A)
have "Lcm (insert x A) = lcm x (Lcm A)" by simp
also have "multiplicity p … = Max (multiplicity p ` insert x A)"
using insert by (subst multiplicity_lcm) (auto simp: Lcm_0_iff)
finally show ?case by simp
qed auto
text ‹
The multiplicity of any prime ‹p› in @{term "Lcm {1..(n::nat)}"} differs from
that in @{term "Lcm {1..(n - 1 :: nat)}"} iff ‹n› is a power of ‹p›, in which case
it is greater by 1.
›
lemma multiplicity_Lcm_atLeast1AtMost_Suc:
fixes p n :: nat
assumes p: "prime p" and n: "n > 0"
shows "multiplicity p (Lcm {1..Suc n}) =
(if ∃k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})"
proof -
define k where "k = Max (multiplicity p ` {1..n})"
define l where "l = multiplicity p (Suc n)"
have eq: "{1..Suc n} = insert (Suc n) {1..n}" by auto
from ‹prime p› have "p > 1" by (auto dest: prime_gt_1_nat)
have "multiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})"
using assms by (subst multiplicity_Lcm) auto
also have "multiplicity p ` {1..Suc n} =
insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})"
by (simp only: eq image_insert)
also have "Max … = max l k"
unfolding l_def k_def using assms by (subst Max.insert) auto
also have "… = (if ∃k. Suc n = p ^ k then 1 else 0) + k"
proof (cases "∃k. Suc n = p ^ k")
case False
have "p ^ l dvd Suc n"
unfolding l_def by (intro multiplicity_dvd)
hence "p ^ l ≤ Suc n"
unfolding l_def by (intro dvd_imp_le multiplicity_dvd) auto
moreover have "Suc n ≠ p ^ l" using False by blast
ultimately have "p ^ l < Suc n" by linarith
moreover have "p ^ l > 0" using ‹p > 1› by (intro zero_less_power) auto
ultimately have "l = multiplicity p (p ^ l)" and "p ^ l ∈ {1..n}"
using ‹prime p› by auto
hence "l ≤ k" unfolding k_def by (intro Max.coboundedI) auto
with False show ?thesis by (simp add: l_def k_def)
next
case True
then obtain x where x: "Suc n = p ^ x" by blast
from x and ‹n > 0› have "x > 0" by (intro Nat.gr0I) auto
from x have [simp]: "l = x" using ‹prime p› by (simp add: l_def)
have "x = k + 1"
proof (intro antisym)
have "p ^ (x - 1) < Suc n"
using ‹x > 0› ‹p > 1› unfolding x by (intro power_strict_increasing) auto
moreover have "p ^ (x - 1) > 0"
using ‹p > 1› by (intro zero_less_power) auto
ultimately have "multiplicity p (p ^ (x - 1)) = x - 1" and "p ^ (x - 1) ∈ {1..n}"
using ‹prime p› by auto
hence "x - 1 ≤ k"
unfolding k_def by (intro Max.coboundedI) force+
thus "x ≤ k + 1" by linarith
next
have "multiplicity p y < x" if "y ∈ {1..n}" for y
proof -
have "p ^ multiplicity p y ≤ y"
using that by (intro dvd_imp_le multiplicity_dvd) auto
also have "… < Suc n" using that by simp
also have "… = p ^ x" by (fact x)
finally show "multiplicity p y < x"
using ‹p > 1› by (subst (asm) power_strict_increasing_iff)
qed
hence "k < x"
using ‹n > 0› unfolding k_def by (subst Max_less_iff) auto
thus "k + 1 ≤ x" by simp
qed
thus ?thesis using True by simp
qed
also have "k = multiplicity p (Lcm {1..n})"
unfolding k_def using ‹n > 0› and ‹prime p› by (subst multiplicity_Lcm) auto
finally show ?thesis .
qed
text ‹
Consequently, \<^term>‹Lcm {1..(n::nat)}› differs from \<^term>‹Lcm {1..(n-1::nat)}›
iff ‹n› is of the form $p^k$ for some prime $p$, in which case it is greater by a factor
of ‹p›.
›
lemma Lcm_atLeast1AtMost_Suc:
"Lcm {1..Suc n} = Lcm {1..n} * (if primepow (Suc n) then aprimedivisor (Suc n) else 1)"
proof (cases "n > 0")
case True
show ?thesis
proof (rule multiplicity_eq_nat)
fix p :: nat assume "prime p"
define x where "x = (if primepow (Suc n) then aprimedivisor (Suc n) else 1)"
have "x > 0"
using ‹n > 0› by (auto simp: x_def intro!: aprimedivisor_pos_nat)
have "multiplicity p (Lcm {1..n} * x) = multiplicity p (Lcm {1..n}) + multiplicity p x"
using ‹prime p› ‹x > 0› by (subst prime_elem_multiplicity_mult_distrib) auto
also consider "∃k. Suc n = p ^ k" | "primepow (Suc n)" "¬(∃k. Suc n = p ^ k)"
| "¬primepow (Suc n)" by blast
hence "multiplicity p x = (if ∃k. Suc n = p ^ k then 1 else 0)"
proof cases
assume "∃k. Suc n = p ^ k"
thus ?thesis using ‹prime p› ‹n > 0›
by (auto simp: x_def aprimedivisor_prime_power intro!: Nat.gr0I)
next
assume *: "primepow (Suc n)" "∄k. Suc n = p ^ k"
then obtain q k where qk: "prime q" "Suc n = q ^ k" "k > 0" "q ≠ p"
by (auto simp: primepow_def)
thus ?thesis using ‹prime p›
by (subst *) (auto simp: x_def aprimedivisor_prime_power prime_multiplicity_other)
next
assume *: "¬primepow (Suc n)"
hence **: "∄k. Suc n = p ^ k" using ‹prime p› ‹n > 0› by auto
from * show ?thesis unfolding x_def
by (subst **) auto
qed
also have "multiplicity p (Lcm {1..n}) + … = multiplicity p (Lcm {1..Suc n})"
using ‹prime p› ‹n > 0› by (subst multiplicity_Lcm_atLeast1AtMost_Suc) (auto simp: x_def)
finally show "multiplicity p (Lcm {1..Suc n}) = multiplicity p (Lcm {1..n} * x)" ..
qed (use ‹n > 0› in ‹auto intro!: Nat.gr0I dest: aprimedivisor_pos_nat›)
qed auto
text ‹
It follows by induction that $\text{Lcm}\ \{1..n\} = e^{\psi(n)}$.
›
lemma Lcm_atLeast1AtMost_conv_ψ:
includes prime_counting_syntax
shows "real (Lcm {1..n}) = exp (ψ (real n))"
proof (induction n)
case (Suc n)
have "real (Lcm {1..Suc n}) =
real (Lcm {1..n}) * (if primepow (Suc n) then aprimedivisor (Suc n) else 1)"
by (subst Lcm_atLeast1AtMost_Suc) auto
also {
assume "primepow (Suc n)"
hence "Suc n > Suc 0" by (rule primepow_gt_Suc_0)
hence "aprimedivisor (Suc n) > 0" by (intro aprimedivisor_pos_nat) auto
}
hence "(if primepow (Suc n) then aprimedivisor (Suc n) else 1) = exp (mangoldt (Suc n))"
by (simp add: mangoldt_def)
also have "Lcm {1..n} * … = exp (ψ (real n + 1))"
using Suc.IH by (simp add: primes_psi_def sum_upto_plus1 exp_add)
finally show ?case by (simp add: add_ac)
qed simp_all
lemma Lcm_upto_real_conv_ψ:
includes prime_counting_syntax
shows "real (Lcm {1..nat ⌊x⌋}) = exp (ψ x)"
by (subst Lcm_atLeast1AtMost_conv_ψ) (simp add: primes_psi_def sum_upto_altdef)
end