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Theorem List for Metamath Proof Explorer - 44001-44100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsprsymrelfo 44001* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃onto𝑅)

Theoremsprsymrelf1o 44002* The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃1-1-onto𝑅)

Theoremsprbisymrel 44003* There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)

Theoremsprsymrelen 44004* The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝑃𝑅)

20.41.11.3  Proper (unordered) pairs

Proper (unordered) pairs are unordered pairs with exactly 2 elements. The set of proper pairs with elements of a class 𝑉 is defined by {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}.

For example, {1, 2} is a proper pair, because 1 ≠ 2 ( see 1ne2 11837). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4755), {1, V} = {1} (see prprc2 4665) or {V, V} = ∅ (see prprc 4666).

Theoremprpair 44005* Characterization of a proper pair: A class is a proper pair iff it consists of exactly two different sets. (Contributed by AV, 11-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝑋𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎𝑏))

Theoremprproropf1olem0 44006 Lemma 0 for prproropf1o 44011. Remark: 𝑂, the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, can alternatively be written as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ (1st𝑥)𝑅(2nd𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))       (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))

Theoremprproropf1olem1 44007* Lemma 1 for prproropf1o 44011. (Contributed by AV, 12-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)

Theoremprproropf1olem2 44008* Lemma 2 for prproropf1o 44011. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑋𝑃) → ⟨inf(𝑋, 𝑉, 𝑅), sup(𝑋, 𝑉, 𝑅)⟩ ∈ 𝑂)

Theoremprproropf1olem3 44009* Lemma 3 for prproropf1o 44011. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)

Theoremprproropf1olem4 44010* Lemma 4 for prproropf1o 44011. (Contributed by AV, 14-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))

Theoremprproropf1o 44011* There is a bijection between the set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)

Theoremprproropen 44012* The set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, are equinumerous. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑉𝑊𝑅 Or 𝑉) → 𝑂𝑃)

Theoremprproropreud 44013* There is exactly one ordered ordered pair fulfilling a wff iff there is exactly one proper pair fulfilling an equivalent wff. (Contributed by AV, 20-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   (𝜑𝑅 Or 𝑉)    &   (𝑥 = ⟨inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)⟩ → (𝜓𝜒))    &   (𝑥 = 𝑧 → (𝜓𝜃))       (𝜑 → (∃!𝑥𝑂 𝜓 ↔ ∃!𝑦𝑃 𝜒))

Theorempairreueq 44014* Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))

Theorempaireqne 44015* Two sets are not equal iff there is exactly one proper pair whose elements are either one of these sets. (Contributed by AV, 27-Jan-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))

20.41.11.4  Set of proper unordered pairs

Syntaxcprpr 44016 Extend class notation with set of proper unordered pairs.
class Pairsproper

Definitiondf-prpr 44017* Define the function which maps a set 𝑣 to the set of proper unordered pairs consisting of exactly two (different) elements of the set 𝑣. (Contributed by AV, 29-Apr-2023.)
Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprval 44018* The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprvalpw 44019* The set of all proper unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprelb 44020 An element of the set of all proper unordered pairs over a given set 𝑉 is a subset of 𝑉 of size two. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (𝑃 ∈ (Pairsproper𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2)))

Theoremprprelprb 44021* A set is an element of the set of all proper unordered pairs over a given set 𝑋 iff it is a pair of different elements of the set 𝑋. (Contributed by AV, 7-May-2023.)
(𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))

Theoremprprspr2 44022* The set of all proper unordered pairs over a given set 𝑉 is the set of all unordered pairs over that set of size two. (Contributed by AV, 29-Apr-2023.)
(Pairsproper𝑉) = {𝑝 ∈ (Pairs‘𝑉) ∣ (♯‘𝑝) = 2}

Theoremprprsprreu 44023* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique unordered pair over 𝑉 of size two fulfilling this wff. (Contributed by AV, 30-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ (Pairs‘𝑉)((♯‘𝑝) = 2 ∧ 𝜑)))

Theoremprprreueq 44024* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique subset of 𝑉 of size two fulfilling this wff. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)))

Theoremsbcpr 44025* The proper substitution of an unordered pair for a setvar variable corresponds to a proper substitution of each of its elements. (Contributed by AV, 7-Apr-2023.)
(𝑝 = {𝑥, 𝑦} → (𝜑𝜓))       ([{𝑎, 𝑏} / 𝑝]𝜑[𝑏 / 𝑦][𝑎 / 𝑥]𝜓)

Theoremreupr 44026* There is a unique unordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 7-Apr-2023.)
(𝑝 = {𝑎, 𝑏} → (𝜓𝜒))    &   (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))       (𝑋𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))

Theoremreuprpr 44027* There is a unique proper unordered pair fulfilling a wff iff there are uniquely two different sets fulfilling a corresponding wff. (Contributed by AV, 30-Apr-2023.)
(𝑝 = {𝑎, 𝑏} → (𝜓𝜒))    &   (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))       (𝑋𝑉 → (∃!𝑝 ∈ (Pairsproper𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝜒 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥𝑦𝜃) → {𝑥, 𝑦} = {𝑎, 𝑏}))))

Theorempoprelb 44028 Equality for unordered pairs with partially ordered elements. (Contributed by AV, 9-Jul-2023.)
(((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorem2exopprim 44029 The existence of an ordered pair fulfilling a wff implies the existence of an unordered pair fulfilling the wff. (Contributed by AV, 29-Jul-2023.)
(∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Theoremreuopreuprim 44030* There is a unique unordered pair with ordered elements fulfilling a wff if there is a unique ordered pair fulfilling the wff. (Contributed by AV, 28-Jul-2023.)
(𝑋𝑉 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑)))

20.41.12  Number theory (extension)

20.41.12.1  Fermat numbers

At first, the (sequence of) Fermat numbers FermatNo (the 𝑛-th Fermat number is denoted as (FermatNo‘𝑛)) is defined, see df-fmtno 44032, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 44081, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 44087. The fourth Fermat number (i.e., the fifth Fermat prime) (FermatNo‘4) = 65537 is currently the biggest number proven to be prime in set.mm, see 65537prm 44080 (previously, it was 4001, see 4001prm 16473).

Another important result of this section is Goldbach's theorem goldbachth 44051, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 44092.

Finally, it is shown that every prime of the form ((2↑𝑘) + 1) must be a Fermat number (i.e., a Fermat prime), see 2pwp1prmfmtno 44094.

Syntaxcfmtno 44031 Extend class notation with the Fermat numbers.
class FermatNo

Definitiondf-fmtno 44032 Define the function that enumerates the Fermat numbers, see definition in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))

Theoremfmtno 44033 The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))

Theoremfmtnoge3 44034 Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ‘3))

Theoremfmtnonn 44035 Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ)

Theoremfmtnom1nn 44036 A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁)))

Theoremfmtnoodd 44037 Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
(𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))

Theoremfmtnorn 44038* A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
(𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)

Theoremfmtnof1 44039 The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.)
FermatNo:ℕ01-1→ℕ

Theoremfmtnoinf 44040 The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
ran FermatNo ∉ Fin

Theoremfmtnorec1 44041 The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1))

Theoremsqrtpwpw2p 44042 The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtnosqrt 44043 The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtno0 44044 The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) = 3

Theoremfmtno1 44045 The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) = 5

Theoremfmtnorec2lem 44046* Lemma for fmtnorec2 44047 (induction step). (Contributed by AV, 29-Jul-2021.)
(𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Theoremfmtnorec2 44047* The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Theoremfmtnodvds 44048 Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 44047, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2))

Theoremgoldbachthlem1 44049 Lemma 1 for goldbachth 44051. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))

Theoremgoldbachthlem2 44050 Lemma 2 for goldbachth 44051. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremgoldbachth 44051 Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑁𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremfmtnorec3 44052* The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛))))

Theoremfmtnorec4 44053 The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2))))

Theoremfmtno2 44054 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17

Theoremfmtno3 44055 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257

Theoremfmtno4 44056 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537

Theoremfmtno5lem1 44057 Lemma 1 for fmtno5 44061. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216

Theoremfmtno5lem2 44058 Lemma 2 for fmtno5 44061. (Contributed by AV, 22-Jul-2021.)
(65536 · 5) = 327680

Theoremfmtno5lem3 44059 Lemma 3 for fmtno5 44061. (Contributed by AV, 22-Jul-2021.)
(65536 · 3) = 196608

Theoremfmtno5lem4 44060 Lemma 4 for fmtno5 44061. (Contributed by AV, 30-Jul-2021.)
(65536↑2) = 4294967296

Theoremfmtno5 44061 The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
(FermatNo‘5) = 4294967297

Theoremfmtno0prm 44062 The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) ∈ ℙ

Theoremfmtno1prm 44063 The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) ∈ ℙ

Theoremfmtno2prm 44064 The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) ∈ ℙ

Theorem257prm 44065 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
257 ∈ ℙ

Theoremfmtno3prm 44066 The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
(FermatNo‘3) ∈ ℙ

Theoremodz2prm2pw 44067 Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.)
(((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

Theoremfmtnoprmfac1lem 44068 Lemma for fmtnoprmfac1 44069: The order of 2 modulo a prime that divides the n-th Fermat number is 2^(n+1). (Contributed by AV, 25-Jul-2021.) (Proof shortened by AV, 18-Mar-2022.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

Theoremfmtnoprmfac1 44069* Divisor of Fermat number (special form of Euler's result, see fmtnofac1 44074): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Theoremfmtnoprmfac2lem1 44070 Lemma for fmtnoprmfac2 44071. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)

Theoremfmtnoprmfac2 44071* Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 44073): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+2)+1 where k is a positive integer. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 2))) + 1))

Theoremfmtnofac2lem 44072* Lemma for fmtnofac2 44073 (Induction step). (Contributed by AV, 30-Jul-2021.)
((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((((𝑁 ∈ (ℤ‘2) ∧ 𝑦 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑦 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) ∧ ((𝑁 ∈ (ℤ‘2) ∧ 𝑧 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑧 = ((𝑘 · (2↑(𝑁 + 2))) + 1))) → ((𝑁 ∈ (ℤ‘2) ∧ (𝑦 · 𝑧) ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 (𝑦 · 𝑧) = ((𝑘 · (2↑(𝑁 + 2))) + 1))))

Theoremfmtnofac2 44073* Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 44074: Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+2)+1 where k is a nonnegative integer. (Contributed by AV, 30-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 2))) + 1))

Theoremfmtnofac1 44074* Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of Fermat Number/Euler's Result", 24-Jul-2021, https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result): "Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+1)+1 where k is a positive integer." Here, however, k must be a nonnegative integer, because k must be 0 to represent 1 (which is a divisor of Fn ).

Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number Fn is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 44073. (Contributed by AV, 30-Jul-2021.)

((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Theoremfmtno4sqrt 44075 The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
(⌊‘(√‘(FermatNo‘4))) = 256

Theoremfmtno4prmfac 44076 If P was a (prime) factor of the fourth Fermat number less than the square root of the fourth Fermat number, it would be either 65 or 129 or 193. (Contributed by AV, 28-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = 65 ∨ 𝑃 = 129 ∨ 𝑃 = 193))

Theoremfmtno4prmfac193 44077 If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = 193)

Theoremfmtno4nprmfac193 44078 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
¬ 193 ∥ (FermatNo‘4)

Theoremfmtno4prm 44079 The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
(FermatNo‘4) ∈ ℙ

Theorem65537prm 44080 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
65537 ∈ ℙ

Theoremfmtnofz04prm 44081 The first five Fermat numbers are prime, see remark in [ApostolNT] p. 7. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ)

Theoremfmtnole4prm 44082 The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ0𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)

Theoremfmtno5faclem1 44083 Lemma 1 for fmtno5fac 44086. (Contributed by AV, 22-Jul-2021.)
(6700417 · 4) = 26801668

Theoremfmtno5faclem2 44084 Lemma 2 for fmtno5fac 44086. (Contributed by AV, 22-Jul-2021.)
(6700417 · 6) = 40202502

Theoremfmtno5faclem3 44085 Lemma 3 for fmtno5fac 44086. (Contributed by AV, 22-Jul-2021.)
(402025020 + 26801668) = 428826688

Theoremfmtno5fac 44086 The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) = (6700417 · 641)

Theoremfmtno5nprm 44087 The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) ∉ ℙ

Theoremprmdvdsfmtnof1lem1 44088* Lemma 1 for prmdvdsfmtnof1 44091. (Contributed by AV, 3-Aug-2021.)
𝐼 = inf({𝑝 ∈ ℙ ∣ 𝑝𝐹}, ℝ, < )    &   𝐽 = inf({𝑝 ∈ ℙ ∣ 𝑝𝐺}, ℝ, < )       ((𝐹 ∈ (ℤ‘2) ∧ 𝐺 ∈ (ℤ‘2)) → (𝐼 = 𝐽 → (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)))

Theoremprmdvdsfmtnof1lem2 44089 Lemma 2 for prmdvdsfmtnof1 44091. (Contributed by AV, 3-Aug-2021.)
((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))

Theoremprmdvdsfmtnof 44090* The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.) (Proof shortened by II, 16-Feb-2023.)
𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝𝑓}, ℝ, < ))       𝐹:ran FermatNo⟶ℙ

Theoremprmdvdsfmtnof1 44091* The mapping of a Fermat number to its smallest prime factor is a one-to-one function. (Contributed by AV, 4-Aug-2021.)
𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝𝑓}, ℝ, < ))       𝐹:ran FermatNo–1-1→ℙ

Theoremprminf2 44092 The set of prime numbers is infinite. The proof of this variant of prminf 16244 is based on Goldbach's theorem goldbachth 44051 (via prmdvdsfmtnof1 44091 and prmdvdsfmtnof1lem2 44089), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties 44089. (Contributed by AV, 4-Aug-2021.)
ℙ ∉ Fin

Theorem2pwp1prm 44093* For every prime number of the form ((2↑𝑘) + 1) 𝑘 must be a power of 2, see Wikipedia "Fermat number", section "Other theorms about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number, 5-Aug-2021. (Contributed by AV, 7-Aug-2021.)
((𝐾 ∈ ℕ ∧ ((2↑𝐾) + 1) ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝐾 = (2↑𝑛))

Theorem2pwp1prmfmtno 44094* Every prime number of the form ((2↑𝑘) + 1) must be a Fermat number. (Contributed by AV, 7-Aug-2021.)
((𝐾 ∈ ℕ ∧ 𝑃 = ((2↑𝐾) + 1) ∧ 𝑃 ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝑃 = (FermatNo‘𝑛))

20.41.12.2  Mersenne primes

"In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2^n-1 for some integer n. They are named after Marin Mersenne ... If n is a composite number then so is 2^n-1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2^p-1 for some prime p.", see Wikipedia "Mersenne prime", 16-Aug-2021, https://en.wikipedia.org/wiki/Mersenne_prime. See also definition in [ApostolNT] p. 4.

This means that if Mn = 2^n-1 is prime, than n must be prime, too, see mersenne 25814. The reverse direction is not generally valid: If p is prime, then Mp = 2^p-1 needs not be prime, e.g. M11 = 2047 = 23 x 89, see m11nprm 44106. This is an example of sgprmdvdsmersenne 44109, stating that if p with p = 3 modulo 4 (here 11) and q=2p+1 (here 23) are prime, then q divides Mp.

"In number theory, a prime number p is a Sophie Germain prime if 2p+1 is also prime. The number 2p+1 associated with a Sophie Germain prime is called a safe prime.", see Wikipedia "Safe and Sophie Germain primes", 21-Aug-2021, https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes 44109. Hence, 11 is a Sophie Germain prime and 2x11+1=23 is its associated safe prime. By sfprmdvdsmersenne 44108, it is shown that if a safe prime q is congruent to 7 modulo 8, then it is a divisor of the Mersenne number with its matching Sophie Germain prime as exponent.

The main result of this section, however, is the formal proof of a theorem of S. Ligh and L. Neal in "A note on Mersenne numbers", see lighneal 44116.

Theoremm2prm 44095 The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑2) − 1) ∈ ℙ

Theoremm3prm 44096 The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑3) − 1) ∈ ℙ

Theoremflsqrt 44097 A condition equivalent to the floor of a square root. (Contributed by AV, 17-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℕ0) → ((⌊‘(√‘𝐴)) = 𝐵 ↔ ((𝐵↑2) ≤ 𝐴𝐴 < ((𝐵 + 1)↑2))))

Theoremflsqrt5 44098 The floor of the square root of a nonnegative number is 5 iff the number is between 25 and 35. (Contributed by AV, 17-Aug-2021.)
((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((25 ≤ 𝑋𝑋 < 36) ↔ (⌊‘(√‘𝑋)) = 5))

Theorem3ndvds4 44099 3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
¬ 3 ∥ 4

Theorem139prmALT 44100 139 is a prime number. In contrast to 139prm 16452, the proof of this theorem uses 3dvds2dec 15677 for checking the divisibility by 3. Although the proof using 3dvds2dec 15677 is longer (regarding size: 1849 characters compared with 1809 for 139prm 16452), the number of essential steps is smaller (301 compared with 327 for 139prm 16452). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
139 ∈ ℙ

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454 45301-45320
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