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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oddz 44801 | An odd number is an integer. (Contributed by AV, 14-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ) | ||
Theorem | evendiv2z 44802 | The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.) |
⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ) | ||
Theorem | oddp1div2z 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) | ||
Theorem | oddm1div2z 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) | ||
Theorem | isodd2 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)) | ||
Theorem | dfodd2 44806 | Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ} | ||
Theorem | dfodd6 44807* | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)} | ||
Theorem | dfeven4 44808* | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} | ||
Theorem | evenm1odd 44809 | The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd ) | ||
Theorem | evenp1odd 44810 | The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd ) | ||
Theorem | oddp1eveni 44811 | The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even ) | ||
Theorem | oddm1eveni 44812 | The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.) |
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even ) | ||
Theorem | evennodd 44813 | An even number is not an odd number. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd ) | ||
Theorem | oddneven 44814 | An odd number is not an even number. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even ) | ||
Theorem | enege 44815 | The negative of an even number is even. (Contributed by AV, 20-Jun-2020.) |
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even ) | ||
Theorem | onego 44816 | The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.) |
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd ) | ||
Theorem | m1expevenALTV 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) | ||
Theorem | m1expoddALTV 44818 | Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.) |
⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1) | ||
Theorem | dfeven2 44819 | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧} | ||
Theorem | dfodd3 44820 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} | ||
Theorem | iseven2 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)) | ||
Theorem | isodd3 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)) | ||
Theorem | 2dvdseven 44823 | 2 divides an even number. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Even → 2 ∥ 𝑍) | ||
Theorem | m2even 44824 | A multiple of 2 is an even number. (Contributed by AV, 5-Jun-2023.) |
⊢ (𝑍 ∈ ℤ → (2 · 𝑍) ∈ Even ) | ||
Theorem | 2ndvdsodd 44825 | 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍) | ||
Theorem | 2dvdsoddp1 44826 | 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 + 1)) | ||
Theorem | 2dvdsoddm1 44827 | 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 − 1)) | ||
Theorem | dfeven3 44828 | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0} | ||
Theorem | dfodd4 44829 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1} | ||
Theorem | dfodd5 44830 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0} | ||
Theorem | zefldiv2ALTV 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)) | ||
Theorem | zofldiv2ALTV 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)) | ||
Theorem | oddflALTV 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)) | ||
Theorem | iseven5 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)) | ||
Theorem | isodd7 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)) | ||
Theorem | dfeven5 44836 | Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2} | ||
Theorem | dfodd7 44837 | Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1} | ||
Theorem | gcd2odd1 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) | ||
Theorem | zneoALTV 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 ) → 𝐴 ≠ 𝐵) | ||
Theorem | zeoALTV 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 )) | ||
Theorem | zeo2ALTV 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 )) | ||
Theorem | nneoALTV 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 )) | ||
Theorem | nneoiALTV 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 ) | ||
Theorem | odd2np1ALTV 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)) | ||
Theorem | oddm1evenALTV 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 )) | ||
Theorem | oddp1evenALTV 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 )) | ||
Theorem | oexpnegALTV 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 ) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | ||
Theorem | oexpnegnz 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 ) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | ||
Theorem | bits0ALTV 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 )) | ||
Theorem | bits0eALTV 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)) | ||
Theorem | bits0oALTV 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)) | ||
Theorem | divgcdoddALTV 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 )) | ||
Theorem | opoeALTV 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 ) | ||
Theorem | opeoALTV 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 ) | ||
Theorem | omoeALTV 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 ) → (𝐴 − 𝐵) ∈ Even ) | ||
Theorem | omeoALTV 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 ) → (𝐴 − 𝐵) ∈ Odd ) | ||
Theorem | oddprmALTV 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 ) | ||
Theorem | 0evenALTV 44858 | 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.) |
⊢ 0 ∈ Even | ||
Theorem | 0noddALTV 44859 | 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.) |
⊢ 0 ∉ Odd | ||
Theorem | 1oddALTV 44860 | 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.) |
⊢ 1 ∈ Odd | ||
Theorem | 1nevenALTV 44861 | 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.) |
⊢ 1 ∉ Even | ||
Theorem | 2evenALTV 44862 | 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.) |
⊢ 2 ∈ Even | ||
Theorem | 2noddALTV 44863 | 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.) |
⊢ 2 ∉ Odd | ||
Theorem | nn0o1gt2ALTV 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 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁)) | ||
Theorem | nnoALTV 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) | ||
Theorem | nn0oALTV 44866 | An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) | ||
Theorem | nn0e 44867 | An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0) | ||
Theorem | nneven 44868 | An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ) | ||
Theorem | nn0onn0exALTV 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 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1)) | ||
Theorem | nn0enn0exALTV 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 ∈ ℕ0 ∧ 𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚)) | ||
Theorem | nnennexALTV 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)) | ||
Theorem | nnpw2evenALTV 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 ) | ||
Theorem | epoo 44873 | The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.) |
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd ) | ||
Theorem | emoo 44874 | The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.) |
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Odd ) | ||
Theorem | epee 44875 | The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.) |
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even ) | ||
Theorem | emee 44876 | The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.) |
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Even ) | ||
Theorem | evensumeven 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 )) | ||
Theorem | 3odd 44878 | 3 is an odd number. (Contributed by AV, 20-Jul-2020.) |
⊢ 3 ∈ Odd | ||
Theorem | 4even 44879 | 4 is an even number. (Contributed by AV, 23-Jul-2020.) |
⊢ 4 ∈ Even | ||
Theorem | 5odd 44880 | 5 is an odd number. (Contributed by AV, 23-Jul-2020.) |
⊢ 5 ∈ Odd | ||
Theorem | 6even 44881 | 6 is an even number. (Contributed by AV, 20-Jul-2020.) |
⊢ 6 ∈ Even | ||
Theorem | 7odd 44882 | 7 is an odd number. (Contributed by AV, 20-Jul-2020.) |
⊢ 7 ∈ Odd | ||
Theorem | 8even 44883 | 8 is an even number. (Contributed by AV, 23-Jul-2020.) |
⊢ 8 ∈ Even | ||
Theorem | evenprm2 44884 | A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.) |
⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2)) | ||
Theorem | oddprmne2 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})) | ||
Theorem | oddprmuzge3 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)) | ||
Theorem | evenltle 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) | ||
Theorem | odd2prm2 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)) | ||
Theorem | even3prm2 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)) | ||
Theorem | mogoldbblem 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)) | ||
Theorem | perfectALTVlem1 44891 | Lemma for perfectALTV 44893. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ 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)) | ||
Theorem | perfectALTVlem2 44892 | Lemma for perfectALTV 44893. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ Odd ) & ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵))) ⇒ ⊢ (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1))) | ||
Theorem | perfectALTV 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))))) | ||
"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. | ||
Syntax | cfppr 44894 | Extend class notation with the Fermat pseudoprimes. |
class FPPr | ||
Definition | df-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))}) | ||
Theorem | fppr 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))}) | ||
Theorem | fpprmod 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)}) | ||
Theorem | fpprel 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))) | ||
Theorem | fpprbasnn 44899 | The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.) |
⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ) | ||
Theorem | fpprnn 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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