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Theorem List for Metamath Proof Explorer - 44201-44300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1oddALTV 44201 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∈ Odd

Theorem1nevenALTV 44202 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∉ Even

Theorem2evenALTV 44203 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∈ Even

Theorem2noddALTV 44204 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∉ Odd

Theoremnn0o1gt2ALTV 44205 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 44206 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 44207 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 44208 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)

Theoremnneven 44209 An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.)
((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)

Theoremnn0onn0exALTV 44210* 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 44211* 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 44212* 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 44213 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 44214 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd )

Theorememoo 44215 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Odd )

Theoremepee 44216 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )

Theorememee 44217 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Even )

Theoremevensumeven 44218 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 44219 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
3 ∈ Odd

Theorem4even 44220 4 is an even number. (Contributed by AV, 23-Jul-2020.)
4 ∈ Even

Theorem5odd 44221 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
5 ∈ Odd

Theorem6even 44222 6 is an even number. (Contributed by AV, 20-Jul-2020.)
6 ∈ Even

Theorem7odd 44223 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
7 ∈ Odd

Theorem8even 44224 8 is an even number. (Contributed by AV, 23-Jul-2020.)
8 ∈ Even

Theoremevenprm2 44225 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
(𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))

Theoremoddprmne2 44226 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 44227 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 44228 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 44229 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 44230 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 44231* Lemma for mogoldbb 44296. (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 44232 Lemma for perfectALTV 44234. (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 44233 Lemma for perfectALTV 44234. (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 44234* 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 16115].

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 44254] 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 44254, 29-May-2023.

Syntaxcfppr 44235 Extend class notation with the Fermat pseudoprimes.
class FPPr

Definitiondf-fppr 44236* 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 44237* 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 44238* 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 44239 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 44240 The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Theoremfpprnn 44241 A Fermat pseudoprime to the base 𝑁 is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ ( FPPr ‘𝑁) → 𝑋 ∈ ℕ)

Theoremfppr2odd 44242 A Fermat pseudoprime to the base 2 is odd. (Contributed by AV, 5-Jun-2023.)
(𝑋 ∈ ( FPPr ‘2) → 𝑋 ∈ Odd )

Theorem11t31e341 44243 341 is the product of 11 and 31. (Contributed by AV, 3-Jun-2023.)
(11 · 31) = 341

Theorem2exp340mod341 44244 Eight to the eighth power modulo nine is one. (Contributed by AV, 3-Jun-2023.)
((2↑340) mod 341) = 1

Theorem341fppr2 44245 341 is the (smallest) Poulet number (Fermat pseudoprime to the base 2). (Contributed by AV, 3-Jun-2023.)
341 ∈ ( FPPr ‘2)

Theorem4fppr1 44246 4 is the (smallest) Fermat pseudoprime to the base 1. (Contributed by AV, 3-Jun-2023.)
4 ∈ ( FPPr ‘1)

Theorem8exp8mod9 44247 Eight to the eighth power modulo nine is one. (Contributed by AV, 2-Jun-2023.)
((8↑8) mod 9) = 1

Theorem9fppr8 44248 9 is the (smallest) Fermat pseudoprime to the base 8. (Contributed by AV, 2-Jun-2023.)
9 ∈ ( FPPr ‘8)

Theoremdfwppr 44249 Alternate definition of a weak pseudoprime 𝑋, which fulfils (𝑁𝑋)≡𝑁 (modulo 𝑋), see Wikipedia "Fermat pseudoprime", https://en.wikipedia.org/wiki/Fermat_pseudoprime, 29-May-2023. (Contributed by AV, 31-May-2023.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℕ) → (((𝑁𝑋) mod 𝑋) = (𝑁 mod 𝑋) ↔ 𝑋 ∥ ((𝑁𝑋) − 𝑁)))

Theoremfpprwppr 44250 A Fermat pseudoprime to the base 𝑁 is a weak pseudoprime (see Wikipedia "Fermat pseudoprime", 29-May-2023, https://en.wikipedia.org/wiki/Fermat_pseudoprime. (Contributed by AV, 31-May-2023.)
(𝑋 ∈ ( FPPr ‘𝑁) → ((𝑁𝑋) mod 𝑋) = (𝑁 mod 𝑋))

Theoremfpprwpprb 44251 An integer 𝑋 which is coprime with an integer 𝑁 is a Fermat pseudoprime to the base 𝑁 iff it is a weak pseudoprime to the base 𝑁. (Contributed by AV, 2-Jun-2023.)
((𝑋 gcd 𝑁) = 1 → (𝑋 ∈ ( FPPr ‘𝑁) ↔ ((𝑋 ∈ (ℤ‘4) ∧ 𝑋 ∉ ℙ) ∧ (𝑁 ∈ ℕ ∧ ((𝑁𝑋) mod 𝑋) = (𝑁 mod 𝑋)))))

Theoremfpprel2 44252 An alternate definition for a Fermat pseudoprime to the base 2. (Contributed by AV, 5-Jun-2023.)
(𝑋 ∈ ( FPPr ‘2) ↔ ((𝑋 ∈ (ℤ‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2))

Theoremnfermltl8rev 44253 Fermat's little theorem with base 8 reversed is not generally true: There is an integer 𝑝 (for example 9, see 9fppr8 44248) so that "𝑝 is prime" does not follow from 8↑𝑝≡8 (mod 𝑝). (Contributed by AV, 3-Jun-2023.)
𝑝 ∈ (ℤ‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)

Theoremnfermltl2rev 44254 Fermat's little theorem with base 2 reversed is not generally true: There is an integer 𝑝 (for example 341, see 341fppr2 44245) so that "𝑝 is prime" does not follow from 2↑𝑝≡2 (mod 𝑝). (Contributed by AV, 3-Jun-2023.)
𝑝 ∈ (ℤ‘3) ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ)

Theoremnfermltlrev 44255* Fermat's little theorem reversed is not generally true: There are integers 𝑎 and 𝑝 so that "𝑝 is prime" does not follow from 𝑎𝑝𝑎 (mod 𝑝). (Contributed by AV, 3-Jun-2023.)
𝑎 ∈ ℤ ∃𝑝 ∈ (ℤ‘3) ¬ (((𝑎𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ)

20.41.14.2  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 44259), "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 44292 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 44292): "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 44260, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternately, df-gbo 44261 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 44259), 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 44282, 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 44321, ax-hgprmladder 44325 and ax-tgoldbachgt 44322.

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 44322
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 44259 A positive even integer that can be expressed as the sum of two odd primes. See https://mathworld.wolfram.com/GoldbachNumber.html 44259
weak odd Goldbach number gbow df-gbow 44260 A positive odd integer that can be expressed as the sum of three primes.
odd Goldbach number strong odd Goldbach number gbo df-gbo 44261 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 44293 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 44293, [*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 44287, sbgoldbwt 44288 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, 44288 [*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 44289 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 44289, https://mathworld.wolfram.com/GoldbachConjecture.html 44289 or section 7.4 in [Helfgott] p. 71.
Goldbach's original conjecture (modern version) the "ternary" Goldbach conjecture mogoldb, m sbgoldbm 44295 Every integer greater than 5 can be written as the sum of three primes. See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture 44295, and https://mathworld.wolfram.com/GoldbachConjecture.html 44295
Goldbach's original conjecture (original version) ogoldb, o sbgoldbo 44298 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 44298, and https://mathworld.wolfram.com/GoldbachConjecture.html 44298

Syntaxcgbe 44256 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
class GoldbachEven

Syntaxcgbow 44257 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 44258 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 44259* 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 44260* 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 44261* 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 44262* 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 44263* 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 44264* 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 44265 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theoremgbowodd 44266 A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Theoremgbogbow 44267 A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW )

Theoremgboodd 44268 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theoremgbepos 44269 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theoremgbowpos 44270 Any weak odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ)

Theoremgbopos 44271 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theoremgbegt5 44272 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theoremgbowgt5 44273 Any weak odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 5 < 𝑍)

Theoremgbowge7 44274 Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 44283, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍)

Theoremgboge9 44275 Any odd Goldbach number is greater than or equal to 9. Because of 9gbo 44285, this bound is strict. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 9 ≤ 𝑍)

Theoremgbege6 44276 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 44282, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theoremgbpart6 44277 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)
6 = (3 + 3)

Theoremgbpart7 44278 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
7 = ((2 + 2) + 3)

Theoremgbpart8 44279 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
8 = (3 + 5)

Theoremgbpart9 44280 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
9 = ((3 + 3) + 3)

Theoremgbpart11 44281 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
11 = ((3 + 3) + 5)

Theorem6gbe 44282 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
6 ∈ GoldbachEven

Theorem7gbow 44283 7 is a weak odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
7 ∈ GoldbachOddW

Theorem8gbe 44284 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
8 ∈ GoldbachEven

Theorem9gbo 44285 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOdd

Theorem11gbo 44286 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOdd

Theoremstgoldbwt 44287 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 44288* 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 44289* 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 44290 Lemma 1 for sbgoldbalt 44292: 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 44291 Lemma 2 for sbgoldbalt 44292: 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 44292* 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 44293* 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 44294* 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 44295* 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 44296* 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 44297* 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 44298* 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))

Theoremnnsum3primes4 44299* 4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))

Theoremnnsum4primes4 44300* 4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))

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