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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgboodd 39001 An odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
GoldbachOdd Odd

Theoremgboagbo 39002 An odd Goldbach number (strong version) is an odd Goldbach number (weak version). (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV GoldbachOdd

Theoremgboaodd 39003 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV Odd

Theoremgbepos 39004 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
GoldbachEven

Theoremgbopos 39005 Any odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
GoldbachOdd

Theoremgboapos 39006 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV

Theoremgbegt5 39007 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
GoldbachEven

Theoremgbogt5 39008 Any odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
GoldbachOdd

Theoremgboge7 39009 Any odd Goldbach number is greater than or equal to 7. Because of 7gbo 39018, this bound is strict. (Contributed by AV, 20-Jul-2020.)
GoldbachOdd

Theoremgboage9 39010 Any odd Goldbach number (strong version) is greater than or equal to 9. Because of 9gboa 39020, this bound is strict. (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV

Theoremgbege6 39011 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 39017, this bound is strict. (Contributed by AV, 20-Jul-2020.)
GoldbachEven

Theoremgbpart6 39012 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)

Theoremgbpart7 39013 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)

Theoremgbpart8 39014 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)

Theoremgbpart9 39015 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)

Theoremgbpart11 39016 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
;

Theorem6gbe 39017 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
GoldbachEven

Theorem7gbo 39018 7 is an odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
GoldbachOdd

Theorem8gbe 39019 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
GoldbachEven

Theorem9gboa 39020 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV

Theorem11gboa 39021 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
; GoldbachOddALTV

Theoremstgoldbwt 39022 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
Odd GoldbachOddALTV Odd GoldbachOdd

Theorembgoldbwt 39023* If the binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.)
Even GoldbachEven Odd GoldbachOdd

Theorembgoldbst 39024* If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
Even GoldbachEven Odd GoldbachOddALTV

Theoremsgoldbaltlem1 39025 Lemma 1 for sgoldbalt 39027: 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.)
Even Odd

Theoremsgoldbaltlem2 39026 Lemma 2 for sgoldbalt 39027: 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.)
Even Odd Odd

Theoremsgoldbalt 39027* 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.)
Even GoldbachEven Even

Theoremnnsum3primes4 39028* 4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)

Theoremnnsum4primes4 39029* 4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)

Theoremnnsum3primesprm 39030* Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.)

Theoremnnsum4primesprm 39031* 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.)

Theoremnnsum3primesgbe 39032* Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
GoldbachEven

Theoremnnsum4primesgbe 39033* 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.)
GoldbachEven

Theoremnnsum3primesle9 39034* 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.)

Theoremnnsum4primesle9 39035* 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.)

Theoremnnsum4primesodd 39036* 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.)
Odd GoldbachOdd Odd

Theoremnnsum4primesoddALTV 39037* 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.)
Odd GoldbachOddALTV Odd

Theoremevengpop3 39038* 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.)
Odd GoldbachOdd Even GoldbachOdd

Theoremevengpoap3 39039* 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 39040* 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 39041* 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 39042* 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 39043* 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 39044* 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 39045 Lemma 1 for bgoldbtbnd 39049: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
Odd ; GoldbachOddALTV

Theorembgoldbtbndlem2 39046* Lemma 2 for bgoldbtbnd 39049. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                     Odd ..^ Even

Theorembgoldbtbndlem3 39047* Lemma 3 for bgoldbtbnd 39049. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                            Odd ..^ Even

Theorembgoldbtbndlem4 39048* Lemma 4 for bgoldbtbnd 39049. (Contributed by AV, 1-Aug-2020.)
;       ;       Even GoldbachEven               RePart       ..^               ;                     ..^ Odd Odd Odd Odd

Theorembgoldbtbnd 39049* 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 39050 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 39051* 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 39050. (Contributed by AV, 3-Aug-2020.)
; Even GoldbachEven

Axiomax-hgprmladder 39052 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 39053 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 39051, ax-hgprmladder 39052 and bgoldbtbnd 39049. (Contributed by AV, 4-Aug-2020.)
Odd ; ; GoldbachOddALTV

Theoremtgoldbachlt 39054* 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 39053. (Contributed by AV, 4-Aug-2020.)
; Odd GoldbachOddALTV

Axiomax-tgoldbachgt 39055* 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 39056 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 39054 and ax-tgoldbachgt 39055. (Contributed by AV, 2-Aug-2020.)
Odd GoldbachOddALTV

21.33.5  Proth's theorem

Theoremmodexp2m1d 39057 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 39058 Lemma for proththd 39059. (Contributed by AV, 4-Jul-2020.)

Theoremproththd 39059* 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 14929), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)

Theorem5tcu2e40 39060 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
;

Theorem3exp4mod41 39061 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
; ;

Theorem41prothprmlem1 39062 Lemma 1 for 41prothprm 39064. (Contributed by AV, 4-Jul-2020.)
;       ;

Theorem41prothprmlem2 39063 Lemma 2 for 41prothprm 39064. (Contributed by AV, 5-Jul-2020.)
;

Theorem41prothprm 39064 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
;

21.33.6  Words over a set (extension)

21.33.6.1  Truncated words

Theoremwrdred1 39065 A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)
Word ..^ Word

Theoremwrdred1hash 39066 The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
Word ..^

21.33.6.2  Last symbol of a word (extension)

Theoremlswn0 39067 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

21.33.6.3  Concatenations with singleton words (extension)

Theoremccatw2s1cl 39068 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
Word ++ ++ Word

21.33.6.4  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 39069 Syntax for the prefix operator.
prefix

Definitiondf-pfx 39070* 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 39071 Value of a prefix. (Contributed by AV, 2-May-2020.)
prefix substr

Theorempfx00 39072 A zero length prefix. (Contributed by AV, 2-May-2020.)
prefix

Theorempfx0 39073 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
prefix

Theorempfxcl 39074 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
Word prefix Word

Theorempfxmpt 39075* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxres 39076 Value of the prefix extractor as restriction. Could replace swrd0val 12831. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxf 39077 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 12837. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxfn 39078 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
Word prefix ..^

Theorempfxlen 39079 Length of a prefix. Could replace swrd0len 12832. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxid 39080 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxrn 39081 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 39082 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12840. (Contributed by AV, 2-May-2020.)
Word prefix

Theorempfxnd 39083 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 39084 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12846. TODO-AV: Really useful? swrd0len0 12846 is only used in wwlknred 25530. (Contributed by AV, 3-May-2020.)
Word prefix

Theoremaddlenrevpfx 39085 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 39086 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 39087 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12849. (Contributed by AV, 3-May-2020.)
Word ..^ prefix

Theorempfxfv0 39088 The first symbol in a prefix of a word. Could replace swrd0fv0 12850. (Contributed by AV, 3-May-2020.)
Word prefix

Theorempfxtrcfv 39089 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12851. (Contributed by AV, 3-May-2020.)
Word ..^ prefix

Theorempfxtrcfv0 39090 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12852. (Contributed by AV, 3-May-2020.)
Word prefix

Theorempfxfvlsw 39091 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12853. (Contributed by AV, 3-May-2020.)
Word lastS prefix

Theorempfxeq 39092* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12854. (Contributed by AV, 4-May-2020.)
Word Word prefix prefix ..^

Theorempfxtrcfvl 39093 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12860. (Contributed by AV, 5-May-2020.)
Word lastS prefix

Theorempfxsuffeqwrdeq 39094 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12863. (Contributed by AV, 5-May-2020.)
Word Word ..^ prefix prefix substr substr

Theorempfxsuff1eqwrdeq 39095 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12864. (Contributed by AV, 6-May-2020.)
Word Word prefix prefix lastS lastS

Theoremdisjwrdpfx 39096* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12865. (Contributed by AV, 6-May-2020.)
Disj Word prefix

Theoremccatpfx 39097 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
Word prefix ++ substr prefix

Theorempfxccat1 39098 Recover the left half of a concatenated word. Could replace swrdccat1 12867. (Contributed by AV, 6-May-2020.)
Word Word ++ prefix

Theorempfx1 39099 A prefix of length 1. (Contributed by AV, 15-May-2020.)
Word prefix

Theorempfx2 39100 A prefix of length 2. (Contributed by AV, 15-May-2020.)
Word prefix

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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 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 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