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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremevengpoap3 38701* 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.)
Odd GoldbachOddALTV ; Even GoldbachOddALTV

Theoremnnsum4primeseven 38702* 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.)
Odd GoldbachOdd Even

Theoremnnsum4primesevenALTV 38703* 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.)
Odd GoldbachOddALTV ; Even

Theoremwtgoldbnnsum4prm 38704* 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.)
Odd GoldbachOdd

Theoremstgoldbnnsum4prm 38705* 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.)
Odd GoldbachOddALTV

Theorembgoldbnnsum3prm 38706* 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.)
Even GoldbachEven

Theorembgoldbtbndlem1 38707 Lemma 1 for bgoldbtbnd 38711: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
Odd ; GoldbachOddALTV

Theorembgoldbtbndlem2 38708* Lemma 2 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                     Odd ..^ Even

Theorembgoldbtbndlem3 38709* Lemma 3 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                            Odd ..^ Even

Theorembgoldbtbndlem4 38710* Lemma 4 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                     ..^ Odd Odd Odd Odd

Theorembgoldbtbnd 38711* If the binary Goldbach conjecture is valid up to an integer , and there is a series ("ladder") of primes with a difference of at most up to an integer , then the strong ternary Goldbach conjecture is valid up to , 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.)
;       ;       Even GoldbachEven               RePart       ..^               ;                     Odd GoldbachOddALTV

Axiomax-bgbltosilva 38712 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.)
Even ; GoldbachEven

Theorembgoldbachlt 38713* The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big ). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 38712. (Contributed by AV, 3-Aug-2020.)
; Even GoldbachEven

Axiomax-hgprmladder 38714 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.)
RePart ; ; ; ..^ ;

Theoremtgblthelfgott 38715 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 38713, ax-hgprmladder 38714 and bgoldbtbnd 38711. (Contributed by AV, 4-Aug-2020.)
Odd ; ; GoldbachOddALTV

Theoremtgoldbachlt 38716* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 38715. (Contributed by AV, 4-Aug-2020.)
; Odd GoldbachOddALTV

Axiomax-tgoldbachgt 38717* The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed ). 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.)
; Odd GoldbachOddALTV

Theoremtgoldbach 38718 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 38716 and ax-tgoldbachgt 38717. (Contributed by AV, 2-Aug-2020.)
Odd GoldbachOddALTV

21.33.5  Proth's theorem

Theoremmodexp2m1d 38719 The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.)

Theoremproththdlem 38720 Lemma for proththd 38721. (Contributed by AV, 4-Jul-2020.)

Theoremproththd 38721* Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 14786), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)

Theorem5tcu2e40 38722 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
;

Theorem3exp4mod41 38723 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
; ;

Theorem41prothprmlem1 38724 Lemma 1 for 41prothprm 38726. (Contributed by AV, 4-Jul-2020.)
;       ;

Theorem41prothprmlem2 38725 Lemma 2 for 41prothprm 38726. (Contributed by AV, 5-Jul-2020.)
;

Theorem41prothprm 38726 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
;

21.33.6  Words over a set (extension)

21.33.6.1  Last symbol of a word (extension)

Theoremlswn0 38727 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( is the last symbol) and invalid cases ( means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
Word lastS

21.33.6.2  Concatenations with singleton words (extension)

Theoremccatw2s1cl 38728 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
Word ++ ++ Word

21.33.6.3  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix), or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

Syntaxcpfx 38729 Syntax for the prefix operator.
prefix

Definitiondf-pfx 38730* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
prefix substr

Theorempfxval 38731 Value of a prefix. (Contributed by AV, 2-May-2020.)
prefix substr

Theorempfx00 38732 A zero length prefix. (Contributed by AV, 2-May-2020.)
prefix

Theorempfx0 38733 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
prefix

Theorempfxcl 38734 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
Word prefix Word

Theorempfxmpt 38735* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxres 38736 Value of the prefix extractor as restriction. Could replace swrd0val 12716. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxf 38737 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 12722. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxfn 38738 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxlen 38739 Length of a prefix. Could replace swrd0len 12717. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxid 38740 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxrn 38741 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxn0 38742 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12725. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxnd 38743 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
Word prefix

Theorempfxlen0 38744 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12731. TODO-AV: Really useful? swrd0len0 12731 is only used in wwlknred 25386. (Contributed by AV, 3-May-2020.)
Word prefix

Theoremaddlenrevpfx 38745 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
Word substr prefix

Theoremaddlenpfx 38746 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
Word prefix substr

Theorempfxfv 38747 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12734. (Contributed by AV, 3-May-2020.)
Word ..^ prefix

Theorempfxfv0 38748 The first symbol in a prefix of a word. Could replace swrd0fv0 12735. (Contributed by AV, 3-May-2020.)
Word prefix

Theorempfxtrcfv 38749 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12736. (Contributed by AV, 3-May-2020.)
Word ..^ prefix

Theorempfxtrcfv0 38750 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12737. (Contributed by AV, 3-May-2020.)
Word prefix

Theorempfxfvlsw 38751 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12738. (Contributed by AV, 3-May-2020.)
Word lastS prefix

Theorempfxeq 38752* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12739. (Contributed by AV, 4-May-2020.)
Word Word prefix prefix ..^

Theorempfxtrcfvl 38753 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12745. (Contributed by AV, 5-May-2020.)
Word lastS prefix

Theorempfxsuffeqwrdeq 38754 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12748. (Contributed by AV, 5-May-2020.)
Word Word ..^ prefix prefix substr substr

Theorempfxsuff1eqwrdeq 38755 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12749. (Contributed by AV, 6-May-2020.)
Word Word prefix prefix lastS lastS

Theoremdisjwrdpfx 38756* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12750. (Contributed by AV, 6-May-2020.)
Disj Word prefix

Theoremccatpfx 38757 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
Word prefix ++ substr prefix

Theorempfxccat1 38758 Recover the left half of a concatenated word. Could replace swrdccat1 12752. (Contributed by AV, 6-May-2020.)
Word Word ++ prefix

Theorempfx1 38759 A prefix of length 1. (Contributed by AV, 15-May-2020.)
Word prefix

Theorempfx2 38760 A prefix of length 2. (Contributed by AV, 15-May-2020.)
Word prefix

Theorempfxswrd 38761 A prefix of a subword. Could replace swrd0swrd 12756. (Contributed by AV, 8-May-2020.)
Word substr prefix substr

Theoremswrdpfx 38762 A subword of a prefix. Could replace swrdswrd0 12757. (Contributed by AV, 8-May-2020.)
Word prefix substr substr

Theorempfxpfx 38763 A prefix of a prefix. Could replace swrd0swrd0 12758. (Contributed by AV, 8-May-2020.)
Word prefix prefix prefix

Theorempfxpfxid 38764 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 12759. (Contributed by AV, 8-May-2020.)
Word prefix prefix prefix

Theorempfxcctswrd 38765 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 12760. (Contributed by AV, 9-May-2020.)
Word prefix ++ substr

Theoremlenpfxcctswrd 38766 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 12761. (Contributed by AV, 9-May-2020.)
Word prefix ++ substr

Theoremlenrevpfxcctswrd 38767 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 12762. (Contributed by AV, 9-May-2020.)
Word substr ++ prefix

Theorempfxlswccat 38768 Reconstruct a nonempty word from its prefix and last symbol. Could replace wrdeqcats1OLD 12769 resp. swrdccatwrd 12763. (Contributed by AV, 9-May-2020.)
Word prefix ++ lastS

Theoremccats1pfxeq 38769 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 12764. (Contributed by AV, 9-May-2020.)
Word Word prefix ++ lastS

Theoremccats1pfxeqrex 38770* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 12774. (Contributed by AV, 9-May-2020.)
Word Word prefix ++

Theorempfxccatin12lem1 38771 Lemma 1 for pfxccatin12 38773. Could replace swrdccatin12lem2b 12781. (Contributed by AV, 9-May-2020.)
..^ ..^ ..^

Theorempfxccatin12lem2 38772 Lemma 2 for pfxccatin12 38773. Could replace swrdccatin12lem2 12784. (Contributed by AV, 9-May-2020.)
Word Word ..^ ..^ ++ substr prefix substr

Theorempfxccatin12 38773 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12786. (Contributed by AV, 9-May-2020.)
Word Word ++ substr substr ++ prefix

Theorempfxccat3 38774 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12787. (Contributed by AV, 10-May-2020.)
Word Word ++ substr substr substr substr ++ prefix

Theorempfxccatpfx1 38775 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
Word Word ++ prefix prefix

Theorempfxccatpfx2 38776 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
Word Word ++ prefix ++ prefix

Theorempfxccat3a 38777 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12789. (Contributed by AV, 10-May-2020.)
Word Word ++ prefix prefix ++ prefix

Theorempfxccatid 38778 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12792. (Contributed by AV, 10-May-2020.)
Word Word ++ prefix

Theoremccats1pfxeqbi 38779 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12793. (Contributed by AV, 10-May-2020.)
Word Word prefix ++ lastS

Theorempfxccatin12d 38780 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12796. (Contributed by AV, 10-May-2020.)
Word Word                      ++ substr substr ++ prefix

Theoremreuccatpfxs1lem 38781* Lemma for reuccatpfxs1 38782. Could replace reuccats1lem 12775. (Contributed by AV, 9-May-2020.)
Word ++ Word prefix ++

Theoremreuccatpfxs1 38782* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 12776. (Contributed by AV, 10-May-2020.)
Word Word ++ prefix

Theoremsplvalpfx 38783 Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
splice prefix ++ ++ substr

Theoremrepswpfx 38784 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
repeatS prefix repeatS

Theoremcshword2 38785 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
Word cyclShift substr ++ prefix

Theorempfxco 38786 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
Word prefix prefix

21.33.7  Auxiliary theorems for graph theory

Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

21.33.7.1  Negated equality and membership - extension

Theoremelnelall 38787 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

21.33.7.2  Subclasses and subsets - extension

Theoremclel5 38788* Alternate definition of class membership: a class is an element of another class iff there is an element of equal to . (Contributed by AV, 13-Nov-2020.)

Theoremdfss7 38789* Alternate definition of subclass relationship: a class is a subclass of another class iff each element of is equal to an element of . (Contributed by AV, 13-Nov-2020.)

21.33.7.3  The empty set - extension

Theoremralnralall 38790* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)

Theoremfalseral0 38791* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)

Theoremralralimp 38792* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)

21.33.7.4  Unordered and ordered pairs - extension

Theoremelpwdifsn 38793 A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)

Theorempr1eqbg 38794 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Theorempr1nebg 38795 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Theoremprelpw 38796 A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.)

Theoremrexdifpr 38797 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)

Theoremissn 38798* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)

Theoremn0snor2el 38799* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)

Theoremopidg 38800 The ordered pair in Kuratowski's representation. Closed form of opid 4142. (Contributed by AV, 18-Sep-2020.)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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