Home Metamath Proof ExplorerTheorem List (p. 213 of 329) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22452) Hilbert Space Explorer (22453-23975) Users' Mathboxes (23976-32860)

Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchto1ub 21201 The function is upper bounded by a linear term. Corollary of chtub 21027. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 21202 The Chebyshev bound, part 2: The function π is eventually upper bounded by a positive constant times . Alternatively stated, the function π is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchto1lb 21203 The function is lower bounded by a linear term. Corollary of chebbnd1 21197. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 21204 The ψ and functions are asymptotic to each other, so is sufficient to prove either or ψ to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpo1ub 21205 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 21206* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 21207* The sum of the von Mangoldt function over is asymptotic to . Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
Λ

Theoremvmadivsumb 21208* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 21209* Lemma for rplogsum 21252. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 21210* Lemma for rplogsum 21252. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 21211 Lemma for dchrisum0lem1 21241. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 21212* Lemma for rpvmasum 21251. Calculate the "trivial case" estimate Λ , where is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                     Λ

Theoremdchrisumlema 21213* Lemma for dchrisum 21217. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 21214* Lemma for dchrisum 21217. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 21215* Lemma for dchrisum 21217. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 21216* Lemma for dchrisum 21217. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 21217* If is a positive decreasing function approaching zero, then the infinite sum is convergent, with the partial sum within of the limit . Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlema 21218* Lemma for dchrmusum 21249 and dchrisumn0 21246. Apply dchrisum 21217 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 21219* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded, provided that . Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem1 21220* An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr                                          Λ

Theoremdchrvmasum2lem 21221* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 21220. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 21222* Combine the results of dchrvmasumlem1 21220 and dchrvmasum2lem 21221 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 21223* Lemma for dchrvmasum 21250. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 21224* Lemma for dchrvmasum 21250. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 21225* Lemma for dchrvmasum 21250 and dchrvmasumif 21228. Apply dchrisum 21217 for the function , which is decreasing above (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem1 21226* Lemma for dchrvmasumif 21228. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 21227* Lemma for dchrvmasum 21250. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 21228* An asymptotic approximation for the sum of Λ conditional on the value of the infinite sum . (We will later show that the case is impossible, and hence establish dchrvmasum 21250.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 21229* The set is the collection of all non-principal Dirichlet characters such that the sum is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fval 21230* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 21231* The function , the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0ff 21232* The function is a real function. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem1 21233* Lemma for dchrisum0flb 21235. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0flblem2 21234* Lemma for dchrisum0flb 21235. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               ..^

Theoremdchrisum0flb 21235* The divisor sum of a real Dirichlet character, is lower bounded by zero everywhere and one at the squares. Equation 9.4.29 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fno1 21236* The sum is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremrpvmasum2 21237* A partial result along the lines of rpvmasum 21251. The sum of the von Mangoldt function over those integers (mod ) is asymptotic to , where is the number of non-principal Dirichlet characters with . Our goal is to show this set is empty. Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                            Unit                            Λ

Theoremdchrisum0re 21238* Suppose is a non-principal Dirichlet character with . Then is a real character. Part of Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lema 21239* Lemma for dchrisum0 21245. Apply dchrisum 21217 for the function . (Contributed by Mario Carneiro, 10-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1b 21240* Lemma for dchrisum0lem1 21241. (Contributed by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem1 21241* Lemma for dchrisum0 21245. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2a 21242* Lemma for dchrisum0 21245. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem2 21243* Lemma for dchrisum0 21245. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0lem3 21244* Lemma for dchrisum0 21245. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0 21245* The sum is nonzero for all non-principal Dirichlet characters (i.e. the assumption is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 21219 and dchrvmasumif 21228. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumn0 21246* The sum is nonzero for all non-principal Dirichlet characters (i.e. the assumption is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 21219 and dchrvmasumif 21228. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlem 21247* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem 21248* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrmusum 21249* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum 21250* The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by , is bounded. Equation 9.4.8 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 12-May-2016.)
ℤ/n       RHom              DChr                                   Λ

Theoremrpvmasum 21251* The sum of the von Mangoldt function over those integers (mod ) is asymptotic to . Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
ℤ/n       RHom              Unit                     Λ

Theoremrplogsum 21252* The sum of over the primes (mod ) is asymptotic to . Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)
ℤ/n       RHom              Unit

Theoremdirith2 21253 Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to . Theorem 9.4.1 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
ℤ/n       RHom              Unit

Theoremdirith 21254* Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to . Theorem 9.4.1 of [Shapiro], p. 375. See http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html for an informal exposition. (Contributed by Mario Carneiro, 12-May-2016.)

13.4.12  The Prime Number Theorem

Theoremmudivsum 21255* Asymptotic formula for . Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsumlem 21256* Lemma for mulogsum 21257. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremmulogsum 21257* Asymptotic formula for . Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

Theoremlogdivsum 21258* Asymptotic analysis of . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem1 21259* Asymptotic formula for , with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremmulog2sumlem2 21260* Lemma for mulog2sum 21262. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremmulog2sumlem3 21261* Lemma for mulog2sum 21262. (Contributed by Mario Carneiro, 13-May-2016.)

Theoremmulog2sum 21262* Asymptotic formula for . Equation 10.2.8 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 19-May-2016.)

Theoremvmalogdivsum2 21263* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremvmalogdivsum 21264* The sum Λ is asymptotic to . Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Λ        Λ Λ

Theorem2vmadivsum 21266* The sum ΛΛ is asymptotic to . (Contributed by Mario Carneiro, 30-May-2016.)
Λ Λ

Theoremlogsqvma 21267* A formula for in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlogsqvma2 21268* The Möbius inverse of logsqvma 21267. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Λ Λ Λ

Theoremlog2sumbnd 21269* Bound on the difference between and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem1 21270* Lemma for selberg 21273. Estimation of the asymptotic part of selberglem3 21272. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremselberglem2 21271* Lemma for selberg 21273. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberglem3 21272* Lemma for selberg 21273. Estimation of the left-hand side of logsqvma2 21268. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremselberg 21273* Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Λ ΛΛ . Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselbergb 21274* Convert eventual boundedness in selberg 21273 to boundedness on . (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ

Theoremselberg2lem 21275* Lemma for selberg2 21276. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Λ ψ

Theoremselberg2 21276* Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
ψ Λ ψ

Theoremselberg2b 21277* Convert eventual boundedness in selberg2 21276 to boundedness on any interval . (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ

Theoremchpdifbndlem1 21278* Lemma for chpdifbnd 21280. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ                             ψ ψ

Theoremchpdifbndlem2 21279* Lemma for chpdifbnd 21280. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ               ψ ψ

Theoremchpdifbnd 21280* A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.)
ψ ψ

Theoremlogdivbnd 21281* A bound on a sum of logs, used in pntlemk 21331. This is not as precise as logdivsum 21258 in its asymptotic behavior, but it is valid for all and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremselberg3lem1 21282* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21276 (written here as Λψ ). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ ψ Λ ψ

Theoremselberg3lem2 21283* Lemma for selberg3 21284. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ Λ ψ

Theoremselberg3 21284* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21276 (written here as Λψ ). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ ψ

Theoremselberg4lem1 21285* Lemma for selberg4 21286. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ Λ ψ

Theoremselberg4 21286* The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form ΛΛΛ; we eliminate one of the nested sums by using the definition of ψ Λ. This statement can thus equivalently be written ψ ΛΛΛ . Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ Λ ψ

Theorempntrval 21287* Define the residual of the second Chebyshev function. The goal is to have , or . (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ        ψ

Theorempntrf 21288 Functionality of the residual. Lemma for pnt 21339. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theorempntrmax 21289* There is a bound on the residual valid for all . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theorempntrsumo1 21290* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd 21291* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd2 21292* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ

Theoremselbergr 21293* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ        Λ

Theoremselberg3r 21294* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ

Theoremselberg4r 21295* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ Λ

Theoremselberg34r 21296* The sum of selberg3r 21294 and selberg4r 21295. (Contributed by Mario Carneiro, 31-May-2016.)
ψ        Λ Λ Λ

Theorempntsval 21297* Define the "Selberg function", whose asymptotic behavior is the content of selberg 21273. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        Λ ψ

Theorempntsf 21298* Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergs 21299* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergsb 21300* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 32801-32860
 Copyright terms: Public domain < Previous  Next >