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Theorem List for Metamath Proof Explorer - 44801-44900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoddz 44801 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
(𝑍 ∈ Odd → 𝑍 ∈ ℤ)
 
Theoremevendiv2z 44802 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)
 
Theoremoddp1div2z 44803 The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)
 
Theoremoddm1div2z 44804 The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Odd → ((𝑍 − 1) / 2) ∈ ℤ)
 
Theoremisodd2 44805 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 − 1) / 2) ∈ ℤ))
 
Theoremdfodd2 44806 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}
 
Theoremdfodd6 44807* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}
 
Theoremdfeven4 44808* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}
 
Theoremevenm1odd 44809 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )
 
Theoremevenp1odd 44810 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )
 
Theoremoddp1eveni 44811 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )
 
Theoremoddm1eveni 44812 The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
(𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )
 
Theoremevennodd 44813 An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )
 
Theoremoddneven 44814 An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )
 
Theoremenege 44815 The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
(𝐴 ∈ Even → -𝐴 ∈ Even )
 
Theoremonego 44816 The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
(𝐴 ∈ Odd → -𝐴 ∈ Odd )
 
Theoremm1expevenALTV 44817 Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
(𝑁 ∈ Even → (-1↑𝑁) = 1)
 
Theoremm1expoddALTV 44818 Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
(𝑁 ∈ Odd → (-1↑𝑁) = -1)
 
20.41.13.2  Alternate definitions using the "divides" relation
 
Theoremdfeven2 44819 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧}
 
Theoremdfodd3 44820 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
 
Theoremiseven2 44821 The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ 2 ∥ 𝑍))
 
Theoremisodd3 44822 The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍))
 
Theorem2dvdseven 44823 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Even → 2 ∥ 𝑍)
 
Theoremm2even 44824 A multiple of 2 is an even number. (Contributed by AV, 5-Jun-2023.)
(𝑍 ∈ ℤ → (2 · 𝑍) ∈ Even )
 
Theorem2ndvdsodd 44825 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)
 
Theorem2dvdsoddp1 44826 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))
 
Theorem2dvdsoddm1 44827 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))
 
20.41.13.3  Alternate definitions using the "modulo" operation
 
Theoremdfeven3 44828 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}
 
Theoremdfodd4 44829 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}
 
Theoremdfodd5 44830 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}
 
Theoremzefldiv2ALTV 44831 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Even → (⌊‘(𝑁 / 2)) = (𝑁 / 2))
 
Theoremzofldiv2ALTV 44832 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))
 
TheoremoddflALTV 44833 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
(𝐾 ∈ Odd → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))
 
20.41.13.4  Alternate definitions using the "gcd" operation
 
Theoremiseven5 44834 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
(𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 2))
 
Theoremisodd7 44835 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
(𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 1))
 
Theoremdfeven5 44836 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2}
 
Theoremdfodd7 44837 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1}
 
Theoremgcd2odd1 44838 The greatest common divisor of an odd number and 2 is 1, i.e., 2 and any odd number are coprime. Remark: The proof using dfodd7 44837 is longer (see proof in comment)! (Contributed by AV, 5-Jun-2023.)
(𝑍 ∈ Odd → (𝑍 gcd 2) = 1)
 
20.41.13.5  Theorems of part 5 revised
 
TheoremzneoALTV 44839 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴𝐵)
 
TheoremzeoALTV 44840 An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
(𝑍 ∈ ℤ → (𝑍 ∈ Even ∨ 𝑍 ∈ Odd ))
 
Theoremzeo2ALTV 44841 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
(𝑍 ∈ ℤ → (𝑍 ∈ Even ↔ ¬ 𝑍 ∈ Odd ))
 
TheoremnneoALTV 44842 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℕ → (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd ))
 
TheoremnneoiALTV 44843 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
𝑁 ∈ ℕ       (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd )
 
20.41.13.6  Theorems of part 6 revised
 
Theoremodd2np1ALTV 44844* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁))
 
Theoremoddm1evenALTV 44845 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 − 1) ∈ Even ))
 
Theoremoddp1evenALTV 44846 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 + 1) ∈ Even ))
 
TheoremoexpnegALTV 44847 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.) (Proof shortened by AV, 10-Jul-2022.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∈ Odd ) → (-𝐴𝑁) = -(𝐴𝑁))
 
Theoremoexpnegnz 44848 The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ Odd ) → (-𝐴𝑁) = -(𝐴𝑁))
 
Theorembits0ALTV 44849 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ 𝑁 ∈ Odd ))
 
Theorembits0eALTV 44850 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ Even → ¬ 0 ∈ (bits‘𝑁))
 
Theorembits0oALTV 44851 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ Odd → 0 ∈ (bits‘𝑁))
 
TheoremdivgcdoddALTV 44852 Either 𝐴 / (𝐴 gcd 𝐵) is odd or 𝐵 / (𝐴 gcd 𝐵) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) ∈ Odd ∨ (𝐵 / (𝐴 gcd 𝐵)) ∈ Odd ))
 
TheoremopoeALTV 44853 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Even )
 
TheoremopeoALTV 44854 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Odd )
 
TheoremomoeALTV 44855 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Even )
 
TheoremomeoALTV 44856 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Odd )
 
TheoremoddprmALTV 44857 A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
(𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd )
 
20.41.13.7  Theorems of AV's mathbox revised
 
Theorem0evenALTV 44858 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∈ Even
 
Theorem0noddALTV 44859 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∉ Odd
 
Theorem1oddALTV 44860 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∈ Odd
 
Theorem1nevenALTV 44861 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∉ Even
 
Theorem2evenALTV 44862 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∈ Even
 
Theorem2noddALTV 44863 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∉ Odd
 
Theoremnn0o1gt2ALTV 44864 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁))
 
TheoremnnoALTV 44865 An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ)
 
Theoremnn0oALTV 44866 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0)
 
Theoremnn0e 44867 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)
 
Theoremnneven 44868 An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.)
((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)
 
Theoremnn0onn0exALTV 44869* For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1))
 
Theoremnn0enn0exALTV 44870* For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚))
 
TheoremnnennexALTV 44871* For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.)
((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚))
 
Theoremnnpw2evenALTV 44872 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
(𝑁 ∈ ℕ → (2↑𝑁) ∈ Even )
 
20.41.13.8  Additional theorems
 
Theoremepoo 44873 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd )
 
Theorememoo 44874 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Odd )
 
Theoremepee 44875 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )
 
Theorememee 44876 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Even )
 
Theoremevensumeven 44877 If a summand is even, the other summand is even iff the sum is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ Even ) → (𝐴 ∈ Even ↔ (𝐴 + 𝐵) ∈ Even ))
 
Theorem3odd 44878 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
3 ∈ Odd
 
Theorem4even 44879 4 is an even number. (Contributed by AV, 23-Jul-2020.)
4 ∈ Even
 
Theorem5odd 44880 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
5 ∈ Odd
 
Theorem6even 44881 6 is an even number. (Contributed by AV, 20-Jul-2020.)
6 ∈ Even
 
Theorem7odd 44882 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
7 ∈ Odd
 
Theorem8even 44883 8 is an even number. (Contributed by AV, 23-Jul-2020.)
8 ∈ Even
 
Theoremevenprm2 44884 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
(𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
 
Theoremoddprmne2 44885 Every prime number not being 2 is an odd prime number. (Contributed by AV, 21-Aug-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) ↔ 𝑃 ∈ (ℙ ∖ {2}))
 
Theoremoddprmuzge3 44886 A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.) (Proof shortened by AV, 21-Aug-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ (ℤ‘3))
 
Theoremevenltle 44887 If an even number is greater than another even number, then it is greater than or equal to the other even number plus 2. (Contributed by AV, 25-Dec-2021.)
((𝑁 ∈ Even ∧ 𝑀 ∈ Even ∧ 𝑀 < 𝑁) → (𝑀 + 2) ≤ 𝑁)
 
Theoremodd2prm2 44888 If an odd number is the sum of two prime numbers, one of the prime numbers must be 2. (Contributed by AV, 26-Dec-2021.)
((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))
 
Theoremeven3prm2 44889 If an even number is the sum of three prime numbers, one of the prime numbers must be 2. (Contributed by AV, 25-Dec-2021.)
((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2))
 
Theoremmogoldbblem 44890* Lemma for mogoldbb 44955. (Contributed by AV, 26-Dec-2021.)
(((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 ∈ Even ∧ (𝑁 + 2) = ((𝑃 + 𝑄) + 𝑅)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑁 = (𝑝 + 𝑞))
 
20.41.13.9  Perfect Number Theorem (revised)
 
TheoremperfectALTVlem1 44891 Lemma for perfectALTV 44893. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 ∈ Odd )    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
 
TheoremperfectALTVlem2 44892 Lemma for perfectALTV 44893. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 ∈ Odd )    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1)))
 
TheoremperfectALTV 44893* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))
 
20.41.14  Number theory (extension 2)
 
20.41.14.1  Fermat pseudoprimes

"In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem ... [which] states that if p is prime and a is coprime to p, then a^(p-1)-1 is divisible by p [see fermltl 16369].

For an integer a > 1, if a composite integer x divides a^(x-1)-1, then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. The false statement [see nfermltl2rev 44913] that all numbers that pass the Fermat primality test for base 2, are prime, is called the Chinese hypothesis.", see Wikipedia "Fermat pseudoprime", https://en.wikipedia.org/wiki/Fermat_pseudoprime 44913, 29-May-2023.

 
Syntaxcfppr 44894 Extend class notation with the Fermat pseudoprimes.
class FPPr
 
Definitiondf-fppr 44895* Define the function that maps a positive integer to the set of Fermat pseudoprimes to the base of this positive integer. Since Fermat pseudoprimes shall be composite (positive) integers, they must be nonprime integers greater than or equal to 4 (we cannot use 𝑥 ∈ ℕ 𝑥 ∉ ℙ because 𝑥 = 1 would fulfil this requirement, but should not be regarded as "composite" integer). (Contributed by AV, 29-May-2023.)
FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
 
Theoremfppr 44896* The set of Fermat pseudoprimes to the base 𝑁. (Contributed by AV, 29-May-2023.)
(𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))})
 
Theoremfpprmod 44897* The set of Fermat pseudoprimes to the base 𝑁, expressed by a modulo operation instead of the divisibility relation. (Contributed by AV, 30-May-2023.)
(𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)})
 
Theoremfpprel 44898 A Fermat pseudoprime to the base 𝑁. (Contributed by AV, 30-May-2023.)
(𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ (𝑋 ∈ (ℤ‘4) ∧ 𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1)))
 
Theoremfpprbasnn 44899 The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)
 
Theoremfpprnn 44900 A Fermat pseudoprime to the base 𝑁 is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ ( FPPr ‘𝑁) → 𝑋 ∈ ℕ)
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