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Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdivgcdoddALTV 38201 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  e.  NN  /\  B  e.  NN )  ->  ( ( A  /  ( A  gcd  B ) )  e. Odd  \/  ( B  /  ( A  gcd  B ) )  e. Odd  )
 )
 
TheoremopoeALTV 38202 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Odd  )  ->  ( A  +  B )  e. Even  )
 
TheoremopeoALTV 38203 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Even  )  ->  ( A  +  B )  e. Odd  )
 
TheoremomoeALTV 38204 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Odd  )  ->  ( A  -  B )  e. Even  )
 
TheoremomeoALTV 38205 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Even  )  ->  ( A  -  B )  e. Odd  )
 
TheoremoddprmALTV 38206 A prime not equal to  2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  N  e. Odd  )
 
21.33.4.7  Theorems of AV's mathbox revised
 
Theorem0evenALTV 38207 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
 |-  0  e. Even
 
Theorem0noddALTV 38208 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
 |-  0  e/ Odd
 
Theorem1oddALTV 38209 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  1  e. Odd
 
Theorem1nevenALTV 38210 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  1  e/ Even
 
Theorem2evenALTV 38211 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  2  e. Even
 
Theorem2noddALTV 38212 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  2  e/ Odd
 
Theoremnn0o1gt2ALTV 38213 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Odd  )  ->  ( N  =  1  \/  2  <  N ) )
 
TheoremnnoALTV 38214 An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  N  e. Odd  )  ->  ( ( N  -  1
 )  /  2 )  e.  NN )
 
Theoremnn0oALTV 38215 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Odd  )  ->  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 
Theoremnn0e 38216 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Even  )  ->  ( N  /  2 )  e.  NN0 )
 
Theoremnn0onn0exALTV 38217* 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  e.  NN0  /\  N  e. Odd  )  ->  E. m  e.  NN0  N  =  ( ( 2  x.  m )  +  1 ) )
 
Theoremnn0enn0exALTV 38218* 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  e.  NN0  /\  N  e. Even  )  ->  E. m  e.  NN0  N  =  ( 2  x.  m ) )
 
Theoremnnpw2evenALTV 38219 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
 |-  ( N  e.  NN  ->  ( 2 ^ N )  e. Even  )
 
21.33.4.8  Additional theorems
 
Theoremepoo 38220 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Odd  )  ->  ( A  +  B )  e. Odd  )
 
Theorememoo 38221 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Odd  )  ->  ( A  -  B )  e. Odd  )
 
Theoremepee 38222 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Even  )  ->  ( A  +  B )  e. Even  )
 
Theorememee 38223 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Even  )  ->  ( A  -  B )  e. Even  )
 
Theoremevensumeven 38224 If a summand is even, the other summand is even iff the sum is even. (Contributed by AV, 21-Jul-2020.)
 |-  (
 ( A  e.  ZZ  /\  B  e. Even  )  ->  ( A  e. Even  <->  ( A  +  B )  e. Even  ) )
 
Theorem3odd 38225 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
 |-  3  e. Odd
 
Theorem4even 38226 4 is an even number. (Contributed by AV, 23-Jul-2020.)
 |-  4  e. Even
 
Theorem5odd 38227 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
 |-  5  e. Odd
 
Theorem6even 38228 6 is an even number. (Contributed by AV, 20-Jul-2020.)
 |-  6  e. Even
 
Theorem7odd 38229 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
 |-  7  e. Odd
 
Theorem8even 38230 8 is an even number. (Contributed by AV, 23-Jul-2020.)
 |-  8  e. Even
 
Theoremoddprmuzge3 38231 A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.)
 |-  (
 ( P  e.  Prime  /\  P  e. Odd  )  ->  P  e.  ( ZZ>= `  3 ) )
 
Theoremevenprm2 38232 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
 |-  ( P  e.  Prime  ->  ( P  e. Even  <->  P  =  2
 ) )
 
21.33.4.9  Perfect Number Theorem (revised)
 
TheoremperfectALTVlem1 38233 Lemma for perfectALTV 38235. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  e. Odd  )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( ( 2 ^ ( A  +  1 )
 )  e.  NN  /\  ( ( 2 ^
 ( A  +  1 ) )  -  1
 )  e.  NN  /\  ( B  /  (
 ( 2 ^ ( A  +  1 )
 )  -  1 ) )  e.  NN )
 )
 
TheoremperfectALTVlem2 38234 Lemma for perfectALTV 38235. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  e. Odd  )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( B  e.  Prime  /\  B  =  ( ( 2 ^
 ( A  +  1 ) )  -  1
 ) ) )
 
TheoremperfectALTV 38235* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer  N is a perfect number (that is, its divisor sum is  2 N) if and only if it is of the form  2 ^ ( p  - 
1 )  x.  (
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  e.  NN  /\  N  e. Even  )  ->  ( ( 1  sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) ) )
 
21.33.4.10  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, because there are are a strong and a weak version of the ternary Goldbach conjecture. The term Goldbach partion 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 38239), "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. sgoldbalt 38272 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." Our definition of "odd Goldbach Numbers", see df-gbo 38240, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternatively, df-gboa 38241 provides a definition allowing for stating the strong ternaty Goldbach conjecture. 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 seems to be 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), providing the still missing official acceptance (usually obtained by a publication in a peer-reviewed journal). 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 strong Goldbach conjecture (see result of [OeSilva] p. ?) resp. about 9 x 10^30 for the weak Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4). Maybe each of the results must be provided as theorem, like 6gbe 38262, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w/m/O3oBPuzhBAAJ , 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 38295, ax-hgprmladder 38297 and ax-tgoldbachgt 38300.

 
Syntaxcgbe 38236 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
 class GoldbachEven
 
Syntaxcgbo 38237 Extend the definition of a class to include the set of odd numbers which can be written as sum of three primes.
 class GoldbachOdd
 
Syntaxcgboa 38238 Extend the definition of a class to include the set of odd numbers which can be written as sum of three odd primes.
 class GoldbachOddALTV
 
Definitiondf-gbe 38239* 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  A. n  e. Even 
( 4  <  n  ->  n  e. GoldbachEven  ). (Contributed by AV, 14-Jun-2020.)
 |- GoldbachEven  =  {
 z  e. Even  |  E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  z  =  ( p  +  q ) ) }
 
Definitiondf-gbo 38240* Define the set of 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  A. m  e. Odd 
( 5  <  m  ->  m  e. GoldbachOdd  ). (Contributed by AV, 14-Jun-2020.)
 |- GoldbachOdd  =  {
 z  e. Odd  |  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  z  =  ( ( p  +  q )  +  r
 ) }
 
Definitiondf-gboa 38241* Define the set of 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  A. m  e. Odd 
( 7  <  m  ->  m  e. GoldbachOddALTV  ). (Contributed by AV, 26-Jul-2020.)
 |- GoldbachOddALTV  =  {
 z  e. Odd  |  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  (
 ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
 )  /\  z  =  ( ( p  +  q )  +  r
 ) ) }
 
Theoremisgbe 38242* The predicate "is an even Goldbach number". An even Goldbach number is an even number having a Goldbach partition, i.e. which can be written as sum of two odd primes. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachEven  <->  ( Z  e. Even  /\ 
 E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  Z  =  ( p  +  q ) ) ) )
 
Theoremisgbo 38243* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd number integer having a Goldbach partition, i.e. which which can be written as sum of three primes. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachOdd  <->  ( Z  e. Odd  /\ 
 E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  Z  =  ( ( p  +  q
 )  +  r ) ) )
 
Theoremisgboa 38244* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd number integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020.)
 |-  ( Z  e. GoldbachOddALTV 
 <->  ( Z  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
 )  /\  Z  =  ( ( p  +  q )  +  r
 ) ) ) )
 
Theoremgbeeven 38245 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
 |-  ( Z  e. GoldbachEven  ->  Z  e. Even  )
 
Theoremgboodd 38246 An odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
 |-  ( Z  e. GoldbachOdd  ->  Z  e. Odd  )
 
Theoremgboagbo 38247 An odd Goldbach number (strong version) is an odd Goldbach number (weak version). (Contributed by AV, 26-Jul-2020.)
 |-  ( Z  e. GoldbachOddALTV  ->  Z  e. GoldbachOdd  )
 
Theoremgboaodd 38248 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
 |-  ( Z  e. GoldbachOddALTV  ->  Z  e. Odd  )
 
Theoremgbepos 38249 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachEven  ->  Z  e.  NN )
 
Theoremgbopos 38250 Any odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachOdd  ->  Z  e.  NN )
 
Theoremgboapos 38251 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
 |-  ( Z  e. GoldbachOddALTV  ->  Z  e.  NN )
 
Theoremgbegt5 38252 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachEven  ->  5  <  Z )
 
Theoremgbogt5 38253 Any odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachOdd  ->  5  <  Z )
 
Theoremgboge7 38254 Any odd Goldbach number is greater than or equal to 7. Because of 7gbo 38263, this bound is strict. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachOdd  ->  7  <_  Z )
 
Theoremgboage9 38255 Any odd Goldbach number (strong version) is greater than or equal to 9. Because of 9gboa 38265, this bound is strict. (Contributed by AV, 26-Jul-2020.)
 |-  ( Z  e. GoldbachOddALTV  ->  9  <_  Z )
 
Theoremgbege6 38256 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 38262, this bound is strict. (Contributed by AV, 20-Jul-2020.)
 |-  ( Z  e. GoldbachEven  ->  6  <_  Z )
 
Theoremgbpart6 38257 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)
 |-  6  =  ( 3  +  3 )
 
Theoremgbpart7 38258 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
 |-  7  =  ( ( 2  +  2 )  +  3 )
 
Theoremgbpart8 38259 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
 |-  8  =  ( 3  +  5 )
 
Theoremgbpart9 38260 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
 |-  9  =  ( ( 3  +  3 )  +  3 )
 
Theoremgbpart11 38261 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
 |- ; 1 1  =  ( ( 3  +  3 )  +  5 )
 
Theorem6gbe 38262 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  6  e. GoldbachEven
 
Theorem7gbo 38263 7 is an odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  7  e. GoldbachOdd
 
Theorem8gbe 38264 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  8  e. GoldbachEven
 
Theorem9gboa 38265 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
 |-  9  e. GoldbachOddALTV
 
Theorem11gboa 38266 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
 |- ; 1 1  e. GoldbachOddALTV
 
Theoremstgoldbwt 38267 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. n  e. Odd  ( 7  <  n  ->  n  e. GoldbachOddALTV  )  ->  A. n  e. Odd  (
 5  <  n  ->  n  e. GoldbachOdd  ) )
 
Theorembgoldbwt 38268* If the binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 )  ->  A. m  e. Odd 
 ( 5  <  m  ->  m  e. GoldbachOdd  ) )
 
Theorembgoldbst 38269* If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 )  ->  A. m  e. Odd 
 ( 7  <  m  ->  m  e. GoldbachOddALTV  ) )
 
Theoremsgoldbaltlem1 38270 Lemma 1 for sgoldbalt 38272: 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  e.  Prime  /\  Q  e.  Prime )  ->  ( ( N  e. Even  /\  4  <  N  /\  N  =  ( P  +  Q ) )  ->  Q  e. Odd  ) )
 
Theoremsgoldbaltlem2 38271 Lemma 2 for sgoldbalt 38272: 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  e.  Prime  /\  Q  e.  Prime )  ->  ( ( N  e. Even  /\  4  <  N  /\  N  =  ( P  +  Q ) )  ->  ( P  e. Odd  /\  Q  e. Odd  ) ) )
 
Theoremsgoldbalt 38272* An alternate (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.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 ) 
 <-> 
 A. n  e. Even  (
 2  <  n  ->  E. p  e.  Prime  E. q  e.  Prime  n  =  ( p  +  q ) ) )
 
Theoremnnsum3primes4 38273* 4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
 |-  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  4  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) )
 
Theoremnnsum4primes4 38274* 4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  4  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) )
 
Theoremnnsum3primesprm 38275* Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.)
 |-  ( P  e.  Prime  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  P  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum4primesprm 38276* 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  e.  Prime  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  P  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum3primesgbe 38277* Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
 |-  ( N  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum4primesgbe 38278* 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  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  N  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum3primesle9 38279* 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  e.  ( ZZ>=
 `  2 )  /\  N  <_  8 )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  3  /\  N  =  sum_
 k  e.  ( 1
 ... d ) ( f `  k ) ) )
 
Theoremnnsum4primesle9 38280* Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  N  <_  8 )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  4  /\  N  =  sum_
 k  e.  ( 1
 ... d ) ( f `  k ) ) )
 
Theoremnnsum4primesodd 38281* If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  6 )  /\  N  e. Odd  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  (
 1 ... 3 ) ( f `  k ) ) )
 
Theoremnnsum4primesoddALTV 38282* If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= `  8
 )  /\  N  e. Odd  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  (
 1 ... 3 ) ( f `  k ) ) )
 
Theoremevengpop3 38283* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  ) 
 ->  E. o  e. GoldbachOdd  N  =  ( o  +  3
 ) ) )
 
Theoremevengpoap3 38284* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
 
Theoremnnsum4primeseven 38285* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 4 ) ) N  =  sum_ k  e.  (
 1 ... 4 ) ( f `  k ) ) )
 
Theoremnnsum4primesevenALTV 38286* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 4
 ) ) N  =  sum_
 k  e.  ( 1
 ... 4 ) ( f `  k ) ) )
 
Theoremwtgoldbnnsum4prm 38287* If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  4  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theoremstgoldbnnsum4prm 38288* If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  A. n  e.  ( ZZ>=
 `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  n  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) ) )
 
Theorembgoldbnnsum3prm 38289* If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)
 |-  ( A. m  e. Even  ( 4  <  m  ->  m  e. GoldbachEven 
 )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  3  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theorembgoldbtbndlem1 38290 Lemma 1 for bgoldbtbnd 38294: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  e.  ( 7 [,); 1 3 ) ) 
 ->  N  e. GoldbachOddALTV  )
 
Theorembgoldbtbndlem2 38291* Lemma 2 for bgoldbtbnd 38294. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  S  =  ( X  -  ( F `  ( I  -  1 ) ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  (
 1..^ D ) ) 
 ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 ) ) ) 
 /\  ( X  -  ( F `  I ) )  <_  4 )  ->  ( S  e. Even  /\  S  <  N  /\  4  <  S ) ) )
 
Theorembgoldbtbndlem3 38292* Lemma 3 for bgoldbtbnd 38294. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   &    |-  S  =  ( X  -  ( F `  I ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  ( 1..^ D ) )  ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 )
 ) )  /\  4  <  S )  ->  ( S  e. Even  /\  S  <  N 
 /\  4  <  S ) ) )
 
Theorembgoldbtbndlem4 38293* Lemma 4 for bgoldbtbnd 38294. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ( ( ph  /\  I  e.  ( 1..^ D ) )  /\  X  e. Odd  )  ->  ( ( X  e.  (
 ( F `  I
 ) [,) ( F `  ( I  +  1
 ) ) )  /\  ( X  -  ( F `  I ) ) 
 <_  4 )  ->  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  (
 ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
 )  /\  X  =  ( ( p  +  q )  +  r
 ) ) ) )
 
Theorembgoldbtbnd 38294* If the binary Goldbach conjecture is valid up to an integer  N, and there is a series ("ladder") of primes with a difference of at most  N up to an integer  M, then the strong ternary Goldbach conjecture is valid up to  M, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ph  ->  A. n  e. Odd  ( ( 7  < 
 n  /\  n  <  M )  ->  n  e. GoldbachOddALTV  )
 )
 
Axiomax-bgbltosilva 38295 The binary Goldbach conjecture is valid for all even numbers less than or equal to 4 x 10^18, see result of [OeSilva] p. ?. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.)
 |-  (
 ( N  e. Even  /\  4  <  N  /\  N  <_  ( 4  x.  ( 10
 ^; 1 8 ) ) )  ->  N  e. GoldbachEven  )
 
Theorembgoldbachlt 38296* The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big  m). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 38295. (Contributed by AV, 3-Aug-2020.)
 |-  E. m  e.  NN  ( ( 4  x.  ( 10 ^; 1 8 ) )  <_  m  /\  A. n  e. Even  (
 ( 4  <  n  /\  n  <  m ) 
 ->  n  e. GoldbachEven  ) )
 
Axiomax-hgprmladder 38297 There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.)
 |-  E. d  e.  ( ZZ>= `  3 ) E. f  e.  (RePart `  d ) ( ( ( f `  0
 )  =  7  /\  ( f `  1
 )  = ; 1 3  /\  (
 f `  d )  =  (; 8 9  x.  ( 10 ^; 2 9 ) ) )  /\  A. i  e.  ( 0..^ d ) ( ( f `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( f `  ( i  +  1 ) )  -  ( f `  i ) )  < 
 ( ( 4  x.  ( 10 ^; 1 8 ) )  -  4 )  /\  4  <  ( ( f `
  ( i  +  1 ) )  -  ( f `  i
 ) ) ) )
 
Theoremtgblthelfgott 38298 The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 38296, ax-hgprmladder 38297 and bgoldbtbnd 38294. (Contributed by AV, 4-Aug-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  <  (; 8 8  x.  ( 10 ^; 2 9 ) ) )  ->  N  e. GoldbachOddALTV  )
 
Theoremtgoldbachlt 38299* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big  m greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 38298. (Contributed by AV, 4-Aug-2020.)
 |-  E. m  e.  NN  ( ( 8  x.  ( 10 ^; 3 0 ) )  <  m  /\  A. n  e. Odd  (
 ( 7  <  n  /\  n  <  m ) 
 ->  n  e. GoldbachOddALTV  ) )
 
Axiomax-tgoldbachgt 38300* The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed  m). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for m = 10^27. Temporarily provided as "axiom". (Contributed by AV, 2-Aug-2020.)
 |-  E. m  e.  NN  ( m  <_  ( 10 ^; 2 7 )  /\  A. n  e. Odd  ( m  <  n  ->  n  e. GoldbachOddALTV  ) )
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