Home Metamath Proof ExplorerTheorem List (p. 203 of 314) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21458) Hilbert Space Explorer (21459-22981) Users' Mathboxes (22982-31321)

Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremleibpisum 20201 The Leibniz formula for . This version of leibpi 20200 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremlog2cnv 20202 Using the Taylor series for arctan , produce a rapidly convergent series for . (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremlog2tlbnd 20203* Bound the error term in the series of log2cnv 20202. (Contributed by Mario Carneiro, 7-Apr-2015.)

13.3.8  The Birthday Problem

Theoremlog2ublem1 20204 Lemma for log2ub 20207. The proof of log2ub 20207, which is simply the evaluation of log2tlbnd 20203 for , takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator (usually a large power of ) and work with closest approximations of the form for some integer instead. It turns out that for our purposes it is sufficient to take , which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremlog2ublem2 20205* Lemma for log2ub 20207. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremlog2ublem3 20206 Lemma for log2ub 20207. In decimal, this is a proof that the first four terms of the series for is less than . (Contributed by Mario Carneiro, 17-Apr-2015.)
;;;;

Theoremlog2ub 20207 is less than . If written in decimal, this is because 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.)
;; ;;

Theorembirthdaylem1 20208* Lemma for birthday 20211. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthdaylem2 20209* For general and , count the fraction of injective functions from to . (Contributed by Mario Carneiro, 7-May-2015.)

Theorembirthdaylem3 20210* For general and , upper-bound the fraction of injective functions from to . (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthday 20211* The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for and , fewer than half of the set of all functions from to are injective. (Contributed by Mario Carneiro, 17-Apr-2015.)
;       ;;

13.3.9  Areas in R^2

Syntaxcarea 20212 Area of regions in the complex plane.
area

Definitiondf-area 20213* Define the area of a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdmarea 20214* The domain of the area function is the set of finitely measurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareambl 20215 The fibers of a measurable region are finitely meaurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareass 20216 A measurable region is a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdfarea 20217* Rewrite df-area 20213 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaf 20218 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareacl 20219 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareage0 20220 The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaval 20221* The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

13.3.10  More miscellaneous converging sequences

Theoremrlimcnp 20222* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremrlimcnp2 20223* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremrlimcnp3 20224* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremxrlimcnp 20225* Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at . Since any limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld       ordTop t

Theoremefrlim 20226* The limit of the sequence is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 20227). (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremdfef2 20227* The limit of the sequence as goes to is . This is another common definition of . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremcxplim 20228* A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)

Theoremsqrlim 20229 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremrlimcxp 20230* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremo1cxp 20231* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2limlem 20232* A linear factor grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2lim 20233* Any power grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim 20234* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim2 20235* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremdivsqrsumlem 20236* Lemma for divsqrsum 20238 and divsqrsum2 20239. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumf 20237* The function used in divsqrsum 20238 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremdivsqrsum 20238* The sum is asymptotic to with a finite limit . (In fact, this limit is .) (Contributed by Mario Carneiro, 9-May-2016.)

Theoremdivsqrsum2 20239* A bound on the distance of the sum from its asymptotic value . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumo1 20240* The sum has the asymptotic expansion , for some . (Contributed by Mario Carneiro, 10-May-2016.)

13.3.11  Inequality of arithmetic and geometric means

Theoremcvxcl 20241* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremscvxcvx 20242* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremjensenlem1 20243* Lemma for jensen 20245. (Contributed by Mario Carneiro, 4-Jun-2016.)
fld g                             fld g        fld g

Theoremjensenlem2 20244* Lemma for jensen 20245. (Contributed by Mario Carneiro, 21-Jun-2015.)
fld g                             fld g        fld g               fld g        fld g fld g        fld g fld g fld g

Theoremjensen 20245* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.)
fld g               fld g fld g fld g fld g fld g fld g

Theoremamgmlem 20246 Lemma for amgm 20247. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpfld                            g fld g

Theoremamgm 20247 Inequality of arithmetic and geometric means. Here g calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements together), and fld g calculates the group sum in the additive group (i.e. the sum of the elements). (Contributed by Mario Carneiro, 20-Jun-2015.)
mulGrpfld       g fld g

13.3.12  Euler-Mascheroni constant

Syntaxcem 20248 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)

Definitiondf-em 20249 Define the Euler-Macheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 20258. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremlogdifbnd 20250 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)

Theorememcllem1 20251* Lemma for emcl 20258. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem2 20252* Lemma for emcl 20258. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem3 20253* Lemma for emcl 20258. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem4 20254* Lemma for emcl 20258. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem5 20255* Lemma for emcl 20258. The partial sums of the series , which is used in the definition df-em 20249, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem6 20256* Lemma for emcl 20258. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem7 20257* Lemma for emcl 20258 and harmonicbnd 20259. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Theorememcl 20258 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd 20259* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremharmonicbnd2 20260* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theorememre 20261 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememgt0 20262 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd3 20263* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmoniclbnd 20264* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicubnd 20265* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicbnd4 20266* The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)

Theoremfsumharmonic 20267* Bound a finite sum based on the harmonic series, where the "strong" bound only applies asymptotically, and there is a "weak" bound for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)

13.4  Basic number theory

13.4.1  Wilson's theorem

Theoremwilthlem1 20268 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 12816, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremwilthlem2 20269* Lemma for wilth 20271: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from to in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except and , and so each pair multiplies to , and and multiply to , so the full product is equal to . Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset of that is closed under inverse (i.e. all pairs are matched up) and contains multiplies to . Given such a set, we take out one element . If there are no such elements, then which forms the base case. Otherwise, is also closed under inverse and contains , so the induction hypothesis says that this equals ; and the remaining two elements are either equal to each other, in which case wilthlem1 20268 gives that or , and we've already excluded the second case, so the product gives ; or and their product is . In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.)

mulGrpfld                            g        g

Theoremwilthlem3 20270* Lemma for wilth 20271. Here we round out the argument of wilthlem2 20269 with the final step of the induction. The induction argument shows that every subset of that is closed under inverse and contains multiplies to , and clearly itself is such a set. Thus the product of all the elements is , and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.)
mulGrpfld

Theoremwilth 20271 Wilson's theorem. A number is prime iff it is greater or equal to and is congruent to , , or alternatively if divides . In this part of the proof we show the relatively simple reverse implication; see wilthlem3 20270 for the forward implication. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

13.4.2  The Fundamental Theorem of Algebra

Theoremftalem1 20272* Lemma for fta 20279: "growth lemma". There exists some such that is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem2 20273* Lemma for fta 20279. There exists some such that has magnitude greater than outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem3 20274* Lemma for fta 20279. There exists a global minimum of the function . The proof uses a circle of radius where is the value coming from ftalem1 20272; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly                     fld

Theoremftalem4 20275* Lemma for fta 20279: Closure of the auxiliary variables for ftalem5 20276. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem5 20276* Lemma for fta 20279: Main proof. We have already shifted the minimum found in ftalem3 20274 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let be the lowest term in the polynomial that is nonzero, and let be a -th root of . Then an evaluation of where is a sufficiently small positive number yields for the first term and for the -th term, and all higher terms are bounded because is small. Thus , in contradiction to our choice of as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem6 20277* Lemma for fta 20279: Discharge the auxiliary variables in ftalem5 20276. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem7 20278* Lemma for fta 20279. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremfta 20279* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. (Contributed by Mario Carneiro, 15-Sep-2014.)
Poly deg

13.4.3  The Basel problem (ζ(2) = π2/6)

Theorembasellem1 20280 Lemma for basel 20289. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem2 20281* Lemma for basel 20289. Show that is a polynomial of degree , and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
Poly deg coeff

Theorembasellem3 20282* Lemma for basel 20289. Using the binomial theorem and de Moivre's formula, we have the identity , so taking imaginary parts yields , where . (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem4 20283* Lemma for basel 20289. By basellem3 20282, the expression goes to zero whenever for some , so this function enumerates distinct roots of a degree- polynomial, which must therefore be all the roots by fta1 19650. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem5 20284* Lemma for basel 20289. Using vieta1 19654, we can calculate the sum of the roots of as the quotient of the top two coefficients, and since the function enumerates the roots, we are left with an equation that sums the function at the different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem6 20285 Lemma for basel 20289. The function goes to zero because it is bounded by . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem7 20286 Lemma for basel 20289. The function for any fixed goes to . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem8 20287* Lemma for basel 20289. The function of partial sums of the inverse squares is bounded below by and above by , obtained by summing the inequality over the roots of the polynomial , and applying the identity basellem5 20284. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem9 20288* Lemma for basel 20289. Since by basellem8 20287 is bounded by two expressions that tend to , must also go to by the squeeze theorem climsqz 12079. But the series is exactly the partial sums of , so it follows that this is also the value of the infinite sum . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasel 20289 The sum of the inverse squares is . This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). (Contributed by Mario Carneiro, 30-Jul-2014.)

13.4.4  Number-theoretical functions

Syntaxccht 20290 Extend class notation with the first Chebyshev function.

Syntaxcvma 20291 Extend class notation with the von Mangoldt function.
Λ

Syntaxcchp 20292 Extend class notation with the second Chebyshev function.
ψ

Syntaxcppi 20293 Extend class notation with the prime Pi function.
π

Syntaxcmu 20294 Extend class notation with the Möbius function.

Syntaxcsgm 20295 Extend class notation with the divisor function.

Definitiondf-cht 20296* Define the first Chebyshev function, which adds up the logarithms of all primes less than . The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Definitiondf-vma 20297* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Definitiondf-chp 20298* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than . (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Definitiondf-ppi 20299 Define the prime π function, which counts the number of primes less than or equal to . (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Definitiondf-mu 20300* Define the Möbius function, which is zero for non-squarefree numbers and is or for squarefree numbers according as to the number of prime divisors of the number is even or odd. (Contributed by Mario Carneiro, 22-Sep-2014.)

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 31301-31321
 Copyright terms: Public domain < Previous  Next >