P. 484 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

bits0oALTV 48301

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 48302

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 48303

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 48304

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 48305

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 48306

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 48307

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 )

21.48.13.7  Theorems of AV's mathbox revised

Theorem

0evenALTV 48308

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

⊢ 0 ∈ Even

Theorem

0noddALTV 48309

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

⊢ 0 ∉ Odd

Theorem

1oddALTV 48310

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

⊢ 1 ∈ Odd

Theorem

1nevenALTV 48311

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

⊢ 1 ∉ Even

Theorem

2evenALTV 48312

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

⊢ 2 ∈ Even

Theorem

2noddALTV 48313

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

⊢ 2 ∉ Odd

Theorem

nn0o1gt2ALTV 48314

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 48315

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 48316

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 48317

An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ₀)

Theorem

nneven 48318

An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.)

⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)

Theorem

nn0onn0exALTV 48319*

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 48320*

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 48321*

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 48322

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 )

21.48.13.8  Additional theorems

Theorem

epoo 48323

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 48324

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 48325

The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)

⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )

Theorem

emee 48326

The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)

⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Even )

Theorem

evensumeven 48327

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 48328

3 is an odd number. (Contributed by AV, 20-Jul-2020.)

⊢ 3 ∈ Odd

Theorem

4even 48329

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

⊢ 4 ∈ Even

Theorem

5odd 48330

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

⊢ 5 ∈ Odd

Theorem

6even 48331

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

⊢ 6 ∈ Even

Theorem

7odd 48332

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

⊢ 7 ∈ Odd

Theorem

8even 48333

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

⊢ 8 ∈ Even

Theorem

evenprm2 48334

A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)

⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))

Theorem

oddprmne2 48335

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 48336

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 48337

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 48338

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 48339

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 48340*

Lemma for mogoldbb 48405. (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))

21.48.13.9  Perfect Number Theorem (revised)

Theorem

perfectALTVlem1 48341

Lemma for perfectALTV 48343. (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 48342

Lemma for perfectALTV 48343. (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 48343*

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)))))

21.48.14  Number theory (extension 2)

21.48.14.1  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 16833].

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 48363] 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 48363, 29-May-2023.

Syntax

cfppr 48344

Extend class notation with the Fermat pseudoprimes.

class FPPr

Definition

df-fppr 48345*

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 48346*

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 48347*

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 48348

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 48349

The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.)

⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Theorem

fpprnn 48350

A Fermat pseudoprime to the base 𝑁 is a positive integer. (Contributed by AV, 30-May-2023.)

⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑋 ∈ ℕ)

Theorem

fppr2odd 48351

A Fermat pseudoprime to the base 2 is odd. (Contributed by AV, 5-Jun-2023.)

⊢ (𝑋 ∈ ( FPPr ‘2) → 𝑋 ∈ Odd )

Theorem

11t31e341 48352

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

⊢ (;11 · ;31) = ;;341

Theorem

2exp340mod341 48353

Eight to the eighth power modulo nine is one. (Contributed by AV, 3-Jun-2023.)

⊢ ((2↑;;340) mod ;;341) = 1

Theorem

341fppr2 48354

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

⊢ ;;341 ∈ ( FPPr ‘2)

Theorem

4fppr1 48355

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

⊢ 4 ∈ ( FPPr ‘1)

Theorem

8exp8mod9 48356

Eight to the eighth power modulo nine is one. (Contributed by AV, 2-Jun-2023.)

⊢ ((8↑8) mod 9) = 1

Theorem

9fppr8 48357

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

⊢ 9 ∈ ( FPPr ‘8)

Theorem

dfwppr 48358

Alternate definition of a weak pseudoprime 𝑋, which fulfils (𝑁↑𝑋)≡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) → (((𝑁↑𝑋) mod 𝑋) = (𝑁 mod 𝑋) ↔ 𝑋 ∥ ((𝑁↑𝑋) − 𝑁)))

Theorem

fpprwppr 48359

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) → ((𝑁↑𝑋) mod 𝑋) = (𝑁 mod 𝑋))

Theorem

fpprwpprb 48360

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 ∧ ((𝑁↑𝑋) mod 𝑋) = (𝑁 mod 𝑋)))))

Theorem

fpprel2 48361

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))

Theorem

nfermltl8rev 48362

Fermat's little theorem with base 8 reversed is not generally true: There is an integer 𝑝 (for example 9, see 9fppr8 48357) 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)

Theorem

nfermltl2rev 48363

Fermat's little theorem with base 2 reversed is not generally true: There is an integer 𝑝 (for example 341, see 341fppr2 48354) 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)

Theorem

nfermltlrev 48364*

Fermat's little theorem reversed is not generally true: There are integers 𝑎 and 𝑝 so that "𝑝 is prime" does not follow from 𝑎↑𝑝≡𝑎 (mod 𝑝). (Contributed by AV, 3-Jun-2023.)

⊢ ∃𝑎 ∈ ℤ ∃𝑝 ∈ (ℤ_≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ)

21.48.14.2  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 48368), "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 48401 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 48401): "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 48369, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternately, df-gbo 48370 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 48368), 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 48391, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w 48391, 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 48430, ax-hgprmladder 48434 and ax-tgoldbachgt 48431.

Summary/glossary:

Term	Synonyms	Label fragment	Definition/Theorem	Remarks
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 48431
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 48368	A positive even integer that can be expressed as the sum of two odd primes.	See https://mathworld.wolfram.com/GoldbachNumber.html 48368
weak odd Goldbach number		gbow	df-gbow 48369	A positive odd integer that can be expressed as the sum of three primes.
odd Goldbach number	strong odd Goldbach number	gbo	df-gbo 48370	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 48402	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 48402, [*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 48396, sbgoldbwt 48397	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, 48397 [*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 48398	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 48398, https://mathworld.wolfram.com/GoldbachConjecture.html 48398 or section 7.4 in [Helfgott] p. 71.
Goldbach's original conjecture (modern version)	the "ternary" Goldbach conjecture	mogoldb, m	sbgoldbm 48404	Every integer greater than 5 can be written as the sum of three primes.	See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture 48404, and https://mathworld.wolfram.com/GoldbachConjecture.html 48404
Goldbach's original conjecture (original version)		ogoldb, o	sbgoldbo 48407	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 48407, and https://mathworld.wolfram.com/GoldbachConjecture.html 48407

Syntax

cgbe 48365

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

class GoldbachEven

Syntax

cgbow 48366

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

Syntax

cgbo 48367

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

Definition

df-gbe 48368*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ). (Contributed by AV, 14-Jun-2020.)

⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

Definition

df-gbow 48369*

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 < 𝑚 → 𝑚 ∈ GoldbachOddW ). (Contributed by AV, 14-Jun-2020.)

⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

Definition

df-gbo 48370*

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 < 𝑚 → 𝑚 ∈ 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))}

Theorem

isgbe 48371*

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))))

Theorem

isgbow 48372*

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)))

Theorem

isgbo 48373*

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))))

Theorem

gbeeven 48374

An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)

⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theorem

gbowodd 48375

A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Theorem

gbogbow 48376

A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW )

Theorem

gboodd 48377

An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theorem

gbepos 48378

Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theorem

gbowpos 48379

Any weak odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ)

Theorem

gbopos 48380

Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theorem

gbegt5 48381

Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theorem

gbowgt5 48382

Any weak odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍)

Theorem

gbowge7 48383

Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 48392, this bound is strict. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍)

Theorem

gboge9 48384

Any odd Goldbach number is greater than or equal to 9. Because of 9gbo 48394, this bound is strict. (Contributed by AV, 26-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOdd → 9 ≤ 𝑍)

Theorem

gbege6 48385

Any even Goldbach number is greater than or equal to 6. Because of 6gbe 48391, this bound is strict. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theorem

gbpart6 48386

The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)

⊢ 6 = (3 + 3)

Theorem

gbpart7 48387

The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)

⊢ 7 = ((2 + 2) + 3)

Theorem

gbpart8 48388

The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)

⊢ 8 = (3 + 5)

Theorem

gbpart9 48389

The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)

⊢ 9 = ((3 + 3) + 3)

Theorem

gbpart11 48390

The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)

⊢ ;11 = ((3 + 3) + 5)

Theorem

6gbe 48391

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

⊢ 6 ∈ GoldbachEven

Theorem

7gbow 48392

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

⊢ 7 ∈ GoldbachOddW

Theorem

8gbe 48393

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

⊢ 8 ∈ GoldbachEven

Theorem

9gbo 48394

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

⊢ 9 ∈ GoldbachOdd

Theorem

11gbo 48395

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

⊢ ;11 ∈ GoldbachOdd

Theorem

stgoldbwt 48396

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 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Theorem

sbgoldbwt 48397*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ))

Theorem

sbgoldbst 48398*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd ))

Theorem

sbgoldbaltlem1 48399

Lemma 1 for sbgoldbalt 48401: 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 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Theorem

sbgoldbaltlem2 48400

Lemma 2 for sbgoldbalt 48401: 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 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))