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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-odd 38901 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
Odd

Theoremiseven 38902 The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
Even

Theoremisodd 38903 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
Odd

Theoremevenz 38904 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
Even

Theoremoddz 38905 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
Odd

Theoremevendiv2z 38906 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
Even

Theoremoddp1div2z 38907 The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
Odd

Theoremzob 38908 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)

Theoremoddm1div2z 38909 The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
Odd

Theoremisodd2 38910 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
Odd

Theoremdfodd2 38911 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
Odd

Theoremdfodd6 38912* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd

Theoremdfeven4 38913* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even

Theoremevenm1odd 38914 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
Even Odd

Theoremevenp1odd 38915 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
Even Odd

Theoremoddp1eveni 38916 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
Odd Even

Theoremoddm1eveni 38917 The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
Odd Even

Theoremevennodd 38918 An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
Even Odd

Theoremoddneven 38919 An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
Odd Even

Theoremenege 38920 The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
Even Even

Theoremonego 38921 The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
Odd Odd

Theoremm1expevenALTV 38922 Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
Even

Theoremm1expoddALTV 38923 Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
Odd

21.33.4.2  Alternate definitions using the "divides" relation

Theoremdfeven2 38924 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even

Theoremdfodd3 38925 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd

Theoremiseven2 38926 The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
Even

Theoremisodd3 38927 The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
Odd

Theorem2dvdseven 38928 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
Even

Theorem2ndvdsodd 38929 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
Odd

Theorem2dvdsoddp1 38930 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
Odd

Theorem2dvdsoddm1 38931 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
Odd

21.33.4.3  Alternate definitions using the "modulo" operation

Theoremdfeven3 38932 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even

Theoremdfodd4 38933 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd

Theoremdfodd5 38934 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd

Theoremzefldiv2ALTV 38935 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
Even

Theoremzofldiv2ALTV 38936 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
Odd

TheoremoddflALTV 38937 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
Odd

21.33.4.4  Alternate definitions using the "gcd" operation

Theoremgcdzeq 38938 A positive integer is equal to its gcd with an integer if and only if divides . Generalization of gcdeq 14599. (Contributed by AV, 1-Jul-2020.)

Theoremiseven5 38939 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
Even

Theoremisodd7 38940 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
Odd

Theoremdfeven5 38941 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
Even

Theoremdfodd7 38942 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
Odd

21.33.4.5  Theorems of part 5 revised

TheoremzneoALTV 38943 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.)
Even Odd

TheoremzeoALTV 38944 An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
Even Odd

Theoremzeo2ALTV 38945 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
Even Odd

TheoremnneoALTV 38946 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
Even Odd

TheoremnneoiALTV 38947 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
Even Odd

21.33.4.6  Theorems of part 6 revised

Theoremodd2np1ALTV 38948* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
Odd

Theoremoddm1evenALTV 38949 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
Odd Even

Theoremoddp1evenALTV 38950 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
Odd Even

TheoremoexpnegALTV 38951 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.)
Odd

Theoremoexpnegnz 38952 The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
Odd

Theorembits0ALTV 38953 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
bits Odd

Theorembits0eALTV 38954 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
Even bits

Theorembits0oALTV 38955 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
Odd bits

TheoremdivgcdoddALTV 38956 Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
Odd Odd

TheoremopoeALTV 38957 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
Odd Odd Even

TheoremopeoALTV 38958 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
Odd Even Odd

TheoremomoeALTV 38959 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
Odd Odd Even

TheoremomeoALTV 38960 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
Odd Even Odd

TheoremoddprmALTV 38961 A prime not equal to is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
Odd

21.33.4.7  Theorems of AV's mathbox revised

Theorem0evenALTV 38962 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
Even

Theorem0noddALTV 38963 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
Odd

Theorem1oddALTV 38964 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
Odd

Theorem1nevenALTV 38965 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
Even

Theorem2evenALTV 38966 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
Even

Theorem2noddALTV 38967 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
Odd

Theoremnn0o1gt2ALTV 38968 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
Odd

TheoremnnoALTV 38969 An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
Odd

Theoremnn0oALTV 38970 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
Odd

Theoremnn0e 38971 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
Even

Theoremnn0onn0exALTV 38972* 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.)
Odd

Theoremnn0enn0exALTV 38973* 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.)
Even

Theoremnnpw2evenALTV 38974 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
Even

Theoremepoo 38975 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
Even Odd Odd

Theorememoo 38976 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
Even Odd Odd

Theoremepee 38977 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
Even Even Even

Theorememee 38978 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
Even Even Even

Theoremevensumeven 38979 If a summand is even, the other summand is even iff the sum is even. (Contributed by AV, 21-Jul-2020.)
Even Even Even

Theorem3odd 38980 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
Odd

Theorem4even 38981 4 is an even number. (Contributed by AV, 23-Jul-2020.)
Even

Theorem5odd 38982 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
Odd

Theorem6even 38983 6 is an even number. (Contributed by AV, 20-Jul-2020.)
Even

Theorem7odd 38984 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
Odd

Theorem8even 38985 8 is an even number. (Contributed by AV, 23-Jul-2020.)
Even

Theoremoddprmuzge3 38986 A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.)
Odd

Theoremevenprm2 38987 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
Even

21.33.4.9  Perfect Number Theorem (revised)

TheoremperfectALTVlem1 38988 Lemma for perfectALTV 38990. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)
Odd

TheoremperfectALTVlem2 38989 Lemma for perfectALTV 38990. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.)
Odd

TheoremperfectALTV 38990* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer is a perfect number (that is, its divisor sum is ) if and only if it is of the form , where is prime (a Mersenne prime). (It follows from this that 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.)
Even

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 38994), "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 39027 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 38995, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternatively, df-gboa 38996 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 39017, 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 39050, ax-hgprmladder 39052 and ax-tgoldbachgt 39055.

Syntaxcgbe 38991 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
GoldbachEven

Syntaxcgbo 38992 Extend the definition of a class to include the set of odd numbers which can be written as sum of three primes.
GoldbachOdd

Syntaxcgboa 38993 Extend the definition of a class to include the set of odd numbers which can be written as sum of three odd primes.
GoldbachOddALTV

Definitiondf-gbe 38994* 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 Even GoldbachEven . (Contributed by AV, 14-Jun-2020.)
GoldbachEven Even Odd Odd

Definitiondf-gbo 38995* 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 Odd GoldbachOdd . (Contributed by AV, 14-Jun-2020.)
GoldbachOdd Odd

Definitiondf-gboa 38996* 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 Odd GoldbachOddALTV . (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV Odd Odd Odd Odd

Theoremisgbe 38997* 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.)
GoldbachEven Even Odd Odd

Theoremisgbo 38998* 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.)
GoldbachOdd Odd

Theoremisgboa 38999* 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.)
GoldbachOddALTV Odd Odd Odd Odd

Theoremgbeeven 39000 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
GoldbachEven Even

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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