P. 486 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

nnoALTV 48501

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 48502

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 48503

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

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

Theorem

nneven 48504

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

⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)

Theorem

nn0onn0exALTV 48505*

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 48506*

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 48507*

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 48508

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 )

21.49.13.8  Additional theorems

Theorem

epoo 48509

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 48510

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 48511

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

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

Theorem

emee 48512

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

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

Theorem

evensumeven 48513

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 48514

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

⊢ 3 ∈ Odd

Theorem

4even 48515

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

⊢ 4 ∈ Even

Theorem

5odd 48516

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

⊢ 5 ∈ Odd

Theorem

6even 48517

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

⊢ 6 ∈ Even

Theorem

7odd 48518

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

⊢ 7 ∈ Odd

Theorem

8even 48519

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

⊢ 8 ∈ Even

Theorem

evenprm2 48520

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

⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))

Theorem

oddprmne2 48521

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 48522

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 48523

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 48524

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 48525

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 48526*

Lemma for mogoldbb 48591. (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))

21.49.13.9  Perfect Number Theorem (revised)

Theorem

perfectALTVlem1 48527

Lemma for perfectALTV 48529. (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 48528

Lemma for perfectALTV 48529. (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 48529*

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)))))

21.49.14  Number theory (extension 2)

21.49.14.1  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 16868].

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 48549] 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 48549, 29-May-2023.

Syntax

cfppr 48530

Extend class notation with the Fermat pseudoprimes.

class FPPr

Definition

df-fppr 48531*

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 48532*

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 48533*

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 48534

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 48535

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

⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Theorem

fpprnn 48536

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

⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑋 ∈ ℕ)

Theorem

fppr2odd 48537

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

⊢ (𝑋 ∈ ( FPPr ‘2) → 𝑋 ∈ Odd )

Theorem

11t31e341 48538

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

⊢ (;11 · ;31) = ;;341

Theorem

2exp340mod341 48539

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

⊢ ((2↑;;340) mod ;;341) = 1

Theorem

341fppr2 48540

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

⊢ ;;341 ∈ ( FPPr ‘2)

Theorem

4fppr1 48541

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

⊢ 4 ∈ ( FPPr ‘1)

Theorem

8exp8mod9 48542

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

⊢ ((8↑8) mod 9) = 1

Theorem

9fppr8 48543

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

⊢ 9 ∈ ( FPPr ‘8)

Theorem

dfwppr 48544

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 48545

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 48546

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 48547

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 48548

Fermat's little theorem with base 8 reversed is not generally true: There is an integer 𝑝 (for example 9, see 9fppr8 48543) 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 48549

Fermat's little theorem with base 2 reversed is not generally true: There is an integer 𝑝 (for example 341, see 341fppr2 48540) 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 48550*

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)

21.49.14.2  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 48554), "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 48587 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 48587): "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 48555, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternately, df-gbo 48556 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 48554), 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 48577, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w 48577, 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 48616, ax-hgprmladder 48620 and ax-tgoldbachgt 48617.

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 48617
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 48554	A positive even integer that can be expressed as the sum of two odd primes.	See https://mathworld.wolfram.com/GoldbachNumber.html 48554
weak odd Goldbach number		gbow	df-gbow 48555	A positive odd integer that can be expressed as the sum of three primes.
odd Goldbach number	strong odd Goldbach number	gbo	df-gbo 48556	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 48588	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 48588, [*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 48582, sbgoldbwt 48583	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, 48583 [*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 48584	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 48584, https://mathworld.wolfram.com/GoldbachConjecture.html 48584 or section 7.4 in [Helfgott] p. 71.
Goldbach's original conjecture (modern version)	the "ternary" Goldbach conjecture	mogoldb, m	sbgoldbm 48590	Every integer greater than 5 can be written as the sum of three primes.	See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture 48590, and https://mathworld.wolfram.com/GoldbachConjecture.html 48590
Goldbach's original conjecture (original version)		ogoldb, o	sbgoldbo 48593	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 48593, and https://mathworld.wolfram.com/GoldbachConjecture.html 48593

Syntax

cgbe 48551

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

class GoldbachEven

Syntax

cgbow 48552

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 48553

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 48554*

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 48555*

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 48556*

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 48557*

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 48558*

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 48559*

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 48560

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

⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theorem

gbowodd 48561

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

⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Theorem

gbogbow 48562

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

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW )

Theorem

gboodd 48563

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

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theorem

gbepos 48564

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

⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theorem

gbowpos 48565

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

⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ)

Theorem

gbopos 48566

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

⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theorem

gbegt5 48567

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

⊢ (𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theorem

gbowgt5 48568

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

⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍)

Theorem

gbowge7 48569

Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 48578, this bound is strict. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍)

Theorem

gboge9 48570

Any odd Goldbach number is greater than or equal to 9. Because of 9gbo 48580, this bound is strict. (Contributed by AV, 26-Jul-2020.)

⊢ (𝑍 ∈ GoldbachOdd → 9 ≤ 𝑍)

Theorem

gbege6 48571

Any even Goldbach number is greater than or equal to 6. Because of 6gbe 48577, this bound is strict. (Contributed by AV, 20-Jul-2020.)

⊢ (𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theorem

gbpart6 48572

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

⊢ 6 = (3 + 3)

Theorem

gbpart7 48573

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

⊢ 7 = ((2 + 2) + 3)

Theorem

gbpart8 48574

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

⊢ 8 = (3 + 5)

Theorem

gbpart9 48575

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

⊢ 9 = ((3 + 3) + 3)

Theorem

gbpart11 48576

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

⊢ ;11 = ((3 + 3) + 5)

Theorem

6gbe 48577

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

⊢ 6 ∈ GoldbachEven

Theorem

7gbow 48578

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

⊢ 7 ∈ GoldbachOddW

Theorem

8gbe 48579

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

⊢ 8 ∈ GoldbachEven

Theorem

9gbo 48580

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

⊢ 9 ∈ GoldbachOdd

Theorem

11gbo 48581

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

⊢ ;11 ∈ GoldbachOdd

Theorem

stgoldbwt 48582

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 48583*

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 48584*

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 48585

Lemma 1 for sbgoldbalt 48587: 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 48586

Lemma 2 for sbgoldbalt 48587: 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 )))

Theorem

sbgoldbalt 48587*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theorem

sbgoldbb 48588*

If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.)

⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theorem

sgoldbeven3prm 48589*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ((𝑁 ∈ Even ∧ 6 ≤ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑁 = ((𝑝 + 𝑞) + 𝑟)))

Theorem

sbgoldbm 48590*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ_≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theorem

mogoldbb 48591*

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)))

Theorem

sbgoldbmb 48592*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ (ℤ_≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theorem

sbgoldbo 48593*

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 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ_≥‘3)∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theorem

nnsum3primes4 48594*

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)(𝑓‘𝑘))

Theorem

nnsum4primes4 48595*

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)(𝑓‘𝑘))

Theorem

nnsum3primesprm 48596*

Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.) (Proof shortened by AV, 17-Apr-2021.)

⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Theorem

nnsum4primesprm 48597*

Every prime is "the sum of at most 4" (actually one - the prime itself) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)

⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Theorem

nnsum3primesgbe 48598*

Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)

⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Theorem

nnsum4primesgbe 48599*

Any even Goldbach number is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)

⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Theorem

nnsum3primesle9 48600*

Every integer greater than 1 and less than or equal to 8 is the sum of at most 3 primes. (Contributed by AV, 2-Aug-2020.)

⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))