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Theorem List for Metamath Proof Explorer - 41901-42000   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.35.8.4  Alternate definitions using the "gcd" operation

Theoremiseven5 41901 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 41902 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 41903 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2}

Theoremdfodd7 41904 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1}

20.35.8.5  Theorems of part 5 revised

TheoremzneoALTV 41905 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 41906 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 41907 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 41908 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 41909 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 )

20.35.8.6  Theorems of part 6 revised

Theoremodd2np1ALTV 41910* 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 41911 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 41912 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 41913 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.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∈ Odd ) → (-𝐴𝑁) = -(𝐴𝑁))

Theoremoexpnegnz 41914 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 41915 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 41916 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 41917 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 41918 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 41919 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 41920 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 41921 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 41922 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 41923 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 )

20.35.8.7  Theorems of AV's mathbox revised

Theorem0evenALTV 41924 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∈ Even

Theorem0noddALTV 41925 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∉ Odd

Theorem1oddALTV 41926 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∈ Odd

Theorem1nevenALTV 41927 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∉ Even

Theorem2evenALTV 41928 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∈ Even

Theorem2noddALTV 41929 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∉ Odd

Theoremnn0o1gt2ALTV 41930 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 41931 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 41932 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 41933 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)

Theoremnn0onn0exALTV 41934* 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 41935* 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))

Theoremnnpw2evenALTV 41936 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 )

Theoremepoo 41937 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd )

Theorememoo 41938 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Odd )

Theoremepee 41939 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )

Theorememee 41940 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Even )

Theoremevensumeven 41941 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 41942 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
3 ∈ Odd

Theorem4even 41943 4 is an even number. (Contributed by AV, 23-Jul-2020.)
4 ∈ Even

Theorem5odd 41944 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
5 ∈ Odd

Theorem6even 41945 6 is an even number. (Contributed by AV, 20-Jul-2020.)
6 ∈ Even

Theorem7odd 41946 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
7 ∈ Odd

Theorem8even 41947 8 is an even number. (Contributed by AV, 23-Jul-2020.)
8 ∈ Even

Theoremevenprm2 41948 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
(𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))

Theoremoddprmne2 41949 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 41950 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 41951 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 41952 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 41953 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 41954* Lemma for mogoldbb 41998. (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))

20.35.8.9  Perfect Number Theorem (revised)

TheoremperfectALTVlem1 41955 Lemma for perfectALTV 41957. (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 41956 Lemma for perfectALTV 41957. (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 41957* 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)))))

20.35.8.10  Goldbach's conjectures

According to Wikipedia ("Goldbach's conjecture", 20-Jul-2020, https://en.wikipedia.org/wiki/Goldbach's_conjecture) "Goldbach's conjecture ... states: Every even integer greater than 2 can be expressed as the sum of two primes." "It is also known as strong, even or binary Goldbach conjecture, to distinguish it from a weaker conjecture, known ... as the _Goldbach's weak conjecture_, the _odd Goldbach conjecture_, or the _ternary Goldbach conjecture_. This weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes.". In the following, the terms "binary Goldbach conjecture" resp. "ternary Goldbach conjecture" will be used (following the terminology used in [Helfgott] p. 2), because there are a strong and a weak version of the ternary Goldbach conjecture. The term _Goldbach partition_ is used for a sum of two resp. three (odd) primes resulting in an even resp. odd number without further specialization.

Using the definition of a _Goldbach number_, which is "a positive even integer that can be expressed as the sum of two odd primes." (see df-gbe 41961), "another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.". 4 is not a Goldbach number, but it is the sum of two primes (2 and 2) nevertheless. sbgoldbalt 41994 shows that both forms are equivalent.

Hint (see Wikipedia, ("Goldbach's weak conjecture", 26-Jul-2020, https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture): "Some state the [weak] conjecture as 'Every odd number greater than 7 can be expressed as the sum of three odd primes.' This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof [see below] covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture." The definition of "weak odd Goldbach numbers", see df-gbow 41962, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternately, df-gbo 41963 provides a definition of "(strong) odd Goldbach numbers" allowing for stating the strong ternary Goldbach conjecture. In literature, the term "Goldbach number" is used for "even Goldbach numbers" (according to definition df-gbe 41961), whereas there seems to be no explicit names and definitions for "odd Goldbach numbers". Since there are more theorems for "strong odd Goldbach numbers", "odd Goldbach numbers" refers to "strong odd Goldbach numbers" in the following. Otherwise, the term "weak odd Goldbach numbers" is explicitly used.

In contrast to the two versions of the binary Goldbach conjecture, the two versions of the ternary Goldbach conjecture are different not only for small numbers, but the strong version excludes cases like a=2+2+b in general, e.g. 23=2+2+19. Therefore, it seems to be more difficult to prove the strong ternary Goldbach conjecture than the weak version, because there are fewer possible partitions available.

Although the binary Goldbach conjecture is not proven yet, the ternary Goldbach conjecture was proven by Harald Helfgott in 2014 (the weak as well as the strong version, see Main theorem in [Helfgott] p. 2). It would be great if this proof can be formalized with Metamath (although it is not in the Metamath 100 list). This section should be a starting point for this.

The main problem will be to provide means to express the results from checking "small" numbers (performed with a computer): numbers up to about 4 x 10^18 for the binary Goldbach conjecture (see section 2 in [OeSilva] p. 2042, called "even Goldbach conjecture" here) resp. about 9 x 10^30 for the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or 8 x 10^26 (see theorem 2.1 in [OeSilva] p. 2057, called "odd Goldbach conjecture" here). Maybe each of the results must be provided as theorem, like 6gbe 41984, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w , this problem could be solved by using a reflective verifier or adding a concept of verification certificates that can be added into the metamath databases as a reference. To sidestep the computation problem for now, the corresponding theorems are temporarily provided as axioms, see ax-bgbltosilva 42023, ax-hgprmladder 42027 and ax-tgoldbachgt 42024.

# Summary/glossary:

TermSynonymsLabel fragment Definition/TheoremRemarks
binary Goldbach partition simply "Goldbach partition" A pair of primes (p,q) that sum to an even integer 2n=p+q See https://mathworld.wolfram.com/GoldbachPartition.html
weak Goldbach partition gbpart A sum of two resp. three primes resulting in an even resp. odd number without further specialization.
Goldbach partition gbpart A sum of two resp. three odd primes resulting in an even resp. odd number without further specialization.
even Goldbach number simply "Goldbach number" gbe df-gbe 41961 A positive even integer that can be expressed as the sum of two odd primes. See https://mathworld.wolfram.com/GoldbachNumber.html
weak odd Goldbach number gbow df-gbow 41962 A positive odd integer that can be expressed as the sum of three primes.
odd Goldbach number strong odd Goldbach number gbo df-gbo 41963 A positive odd integer that can be expressed as the sum of three odd primes.
strong binary Goldbach conjecture "the" Goldbach conjecture" [*1], even Goldbach conjecture [*2] sbgoldb Every even integer greater than 4 can be expressed as the sum of two odd primes. [*1] Equation (1) in [ApostolNT] p. 304 or [*2] introduction of [OeSilva] p. 2033.
binary Goldbach conjecture[*1][*3] strong Goldbach conjecture [*1], even Goldbach conjecture [*1], or simply "the Goldbach conjecture" [*1][*2] bgoldb, b sbgoldbb 41995 Every even integer greater than 2 can be expressed as the sum of two primes. See [*1] https://en.wikipedia.org/wiki/Goldbach's_conjecture, [*2] statement in [ApostolNT] p. 9 or [*3] section 1.1 in [Helfgott] p. 2.
weak ternary Goldbach conjecture Goldbach's weak conjecture [*1], odd Goldbach conjecture [*1][*3], ternary Goldbach conjecture [*2], ternary Goldbach problem[*1], three-primes problem [*1][*2] wtgoldb, wt stgoldbwt 41989, sbgoldbwt 41990 Every odd number greater than 5 can be expressed as the sum of three primes. See [*1] https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, [*2] section 1.1 in [Helfgott] p. 2 or [*3] section 2.4 in [OeSilva] p. 2057.
ternary Goldbach conjecture strong ternary Goldbach conjecture, the "weak" Goldbach conjecture tgoldb, stgoldb, st sbgoldbst 41991 Every odd number greater than 7 can be expressed as the sum of three odd primes. See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, https://mathworld.wolfram.com/GoldbachConjecture.html or section 7.4 in [Helfgott] p. 71.
Goldbach's original conjecture (modern version) the "ternary" Goldbach conjecture mogoldb, m sbgoldbm 41997 Every integer greater than 5 can be written as the sum of three primes. See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, and https://mathworld.wolfram.com/GoldbachConjecture.html
Goldbach's original conjecture (original version) ogoldb, o sbgoldbo 42000 Every integer greater than 2 can be written as the sum of three "primes" (considered the number 1 to be a "prime"). See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, and https://mathworld.wolfram.com/GoldbachConjecture.html

Syntaxcgbe 41958 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
class GoldbachEven

Syntaxcgbow 41959 Extend the definition of a class to include the set of odd numbers which can be written as a sum of three primes.
class GoldbachOddW

Syntaxcgbo 41960 Extend the definition of a class to include the set of odd numbers which can be written as a sum of three odd primes.
class GoldbachOdd

Definitiondf-gbe 41961* Define the set of (even) Goldbach numbers, which are positive even integers that can be expressed as the sum of two odd primes. By this definition, the binary Goldbach conjecture can be expressed as 𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ). (Contributed by AV, 14-Jun-2020.)
GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

Definitiondf-gbow 41962* Define the set of weak odd Goldbach numbers, which are positive odd integers that can be expressed as the sum of three primes. By this definition, the weak ternary Goldbach conjecture can be expressed as 𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ). (Contributed by AV, 14-Jun-2020.)
GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

Definitiondf-gbo 41963* Define the set of (strong) odd Goldbach numbers, which are positive odd integers that can be expressed as the sum of three odd primes. By this definition, the strong ternary Goldbach conjecture can be expressed as 𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOdd ). (Contributed by AV, 26-Jul-2020.)
GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

Theoremisgbe 41964* The predicate "is an even Goldbach number". An even Goldbach number is an even integer having a Goldbach partition, i.e. which can be written as a sum of two odd primes. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))

Theoremisgbow 41965* The predicate "is a weak odd Goldbach number". A weak odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as a sum of three primes. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Theoremisgbo 41966* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟))))

Theoremgbeeven 41967 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theoremgbowodd 41968 A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Theoremgbogbow 41969 A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW )

Theoremgboodd 41970 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theoremgbepos 41971 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theoremgbowpos 41972 Any weak odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ)

Theoremgbopos 41973 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theoremgbegt5 41974 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theoremgbowgt5 41975 Any weak odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 5 < 𝑍)

Theoremgbowge7 41976 Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 41985, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍)

Theoremgboge9 41977 Any odd Goldbach number is greater than or equal to 9. Because of 9gbo 41987, this bound is strict. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 9 ≤ 𝑍)

Theoremgbege6 41978 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 41984, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theoremgbpart6 41979 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)
6 = (3 + 3)

Theoremgbpart7 41980 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
7 = ((2 + 2) + 3)

Theoremgbpart8 41981 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
8 = (3 + 5)

Theoremgbpart9 41982 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
9 = ((3 + 3) + 3)

Theoremgbpart11 41983 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
11 = ((3 + 3) + 5)

Theorem6gbe 41984 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
6 ∈ GoldbachEven

Theorem7gbow 41985 7 is a weak odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
7 ∈ GoldbachOddW

Theorem8gbe 41986 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
8 ∈ GoldbachEven

Theorem9gbo 41987 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOdd

Theorem11gbo 41988 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOdd

Theoremstgoldbwt 41989 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
(∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))

Theoremsbgoldbwt 41990* If the strong binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ))

Theoremsbgoldbst 41991* If the strong binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOdd ))

Theoremsbgoldbaltlem1 41992 Lemma 1 for sbgoldbalt 41994: If an even number greater than 4 is the sum of two primes, one of the prime summands must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Theoremsbgoldbaltlem2 41993 Lemma 2 for sbgoldbalt 41994: If an even number greater than 4 is the sum of two primes, the primes must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))

Theoremsbgoldbalt 41994* An alternate (related to the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremsbgoldbb 41995* If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremsgoldbeven3prm 41996* If the binary Goldbach conjecture is valid, then an even integer greater than 5 can be expressed as the sum of three primes: Since (𝑁 − 2) is even iff 𝑁 is even, there would be primes 𝑝 and 𝑞 with (𝑁 − 2) = (𝑝 + 𝑞), and therefore 𝑁 = ((𝑝 + 𝑞) + 2). (Contributed by AV, 24-Dec-2021.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ((𝑁 ∈ Even ∧ 6 ≤ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑁 = ((𝑝 + 𝑞) + 𝑟)))

Theoremsbgoldbm 41997* If the strong binary Goldbach conjecture is valid, the modern version of the original formulation of the Goldbach conjecture also holds: Every integer greater than 5 can be expressed as the sum of three primes. (Contributed by AV, 24-Dec-2021.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theoremmogoldbb 41998* If the modern version of the original formulation of the Goldbach conjecture is valid, the (weak) binary Goldbach conjecture also holds. (Contributed by AV, 26-Dec-2021.)
(∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremsbgoldbmb 41999* The strong binary Goldbach conjecture and the modern version of the original formulation of the Goldbach conjecture are equivalent. (Contributed by AV, 26-Dec-2021.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theoremsbgoldbo 42000* If the strong binary Goldbach conjecture is valid, the original formulation of the Goldbach conjecture also holds: Every integer greater than 2 can be expressed as the sum of three "primes" with regarding 1 to be a prime (as Goldbach did). Original text: "Es scheint wenigstens, dass eine jede Zahl, die groesser ist als 2, ein aggregatum trium numerorum primorum sey." (Goldbach, 1742). (Contributed by AV, 25-Dec-2021.)
𝑃 = ({1} ∪ ℙ)       (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘3)∃𝑝𝑃𝑞𝑃𝑟𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))

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