Home Metamath Proof ExplorerTheorem List (p. 132 of 327) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22413) Hilbert Space Explorer (22414-23936) Users' Mathboxes (23937-32689)

Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisprm6 13101* A number is prime iff it satisfies Euclid's lemma euclemma 13100. (Contributed by Mario Carneiro, 6-Sep-2015.)

Theoremexprmfct 13102* Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)

Theoremnprmdvds1 13103 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)

Theoremisprm5 13104* One need only check prime divisors of up to in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmaxprmfct 13105* The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremprmdvdsexp 13106 A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)

Theoremprmdvdsexpb 13107 A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.)

Theoremprmdvdsexpr 13108 If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremprmexpb 13109 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremprmfac1 13110 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)

Theoremdivgcdodd 13111 Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremrpexp 13112 If two numbers and are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremrpexp1i 13113 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)

Theoremrpexp12i 13114 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)

Theoremrpmul 13115 If is relatively prime to and to , it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)

Theoremrpdvds 13116 If is relatively prime to then it is also relatively prime to any divisor of . (Contributed by Mario Carneiro, 19-Jun-2015.)

6.2.2  Properties of the canonical representation of a rational

Syntaxcnumer 13117 Extend class notation to include canonical numerator function.
numer

Syntaxcdenom 13118 Extend class notation to include canonical denominator function.
denom

Definitiondf-numer 13119* The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Definitiondf-denom 13120* The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumval 13121* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremqdenval 13122* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumdencl 13123 Lemma for qnumcl 13124 and qdencl 13125. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqnumcl 13124 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremqdencl 13125 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremfnum 13126 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremfden 13127 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumdenbi 13128 Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqnumdencoprm 13129 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqeqnumdivden 13130 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqmuldeneqnum 13131 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom numer

Theoremdivnumden 13132 Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremdivdenle 13133 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumgt0 13134 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer

Theoremqgt0numnn 13135 A rational is positive iff its canonical numerator is a natural number. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer

Theoremnn0gcdsq 13136 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremzgcdsq 13137 nn0gcdsq 13136 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremnumdensq 13138 Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer numer denom denom

Theoremnumsq 13139 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer numer

Theoremdensq 13140 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
denom denom

Theoremqden1elz 13141 A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
denom

Theoremzsqrelqelz 13142 If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremnonsq 13143 Any integer strictly between two adjacent squares has an irrational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)

6.2.3  Euler's theorem

Syntaxcodz 13144 Extend class notation with the order function on the class of integers mod N.

Syntaxcphi 13145 Extend class notation with the Euler phi function.

Definitiondf-odz 13146* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.)

Definitiondf-phi 13147* Define the Euler phi function, which counts the number of integers less than and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphival 13148* Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicl2 13149 Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicl 13150 Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremphibndlem 13151* Lemma for phibnd 13152. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphibnd 13152 A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicld 13153 Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremphi1 13154 Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdfphi2 13155* Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
..^

Theoremhashdvds 13156* The number of numbers in a given residue class in a finite set of integers. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Theoremphiprmpw 13157 Value of the Euler function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremphiprm 13158 Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremcrt 13159* The Chinese Remainder Theorem: the function that maps to its remainder classes and is 1-1 and onto when and are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
..^        ..^ ..^

Theoremphimullem 13160* Lemma for phimul 13161. (Contributed by Mario Carneiro, 24-Feb-2014.)
..^        ..^ ..^                     ..^        ..^

Theoremphimul 13161 The Euler function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremeulerthlem1 13162* Lemma for eulerth 13164. (Contributed by Mario Carneiro, 8-May-2015.)
..^

Theoremeulerthlem2 13163* Lemma for eulerth 13164. (Contributed by Mario Carneiro, 28-Feb-2014.)
..^

Theoremeulerth 13164 Euler's theorem, a generalization of Fermat's little theorem. If and are coprime, then , mod . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremfermltl 13165 Fermat's little theorem. When is prime, , mod for any . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremprmdiv 13166 Show an explicit expression for the modular inverse of . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdiveq 13167 The modular inverse of is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdivdiv 13168 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremodzval 13169* Value of the order function. This is a function of functions; the inner argument selects the base (i.e. mod for some , often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod . In order to ensure the supremum is well-defined, we only define the expression when and are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremodzcllem 13170 - Lemma for odzcl 13171, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzcl 13171 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzid 13172 Any element raised to the power of its order is . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzdvds 13173 The only powers of that are congruent to are the multiples of the order of . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzphi 13174 The order of any group element is a divisor of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.)

6.2.4  Pythagorean Triples

Theoremcoprimeprodsq 13175 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcoprimeprodsq2 13176 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopoe 13177 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 13178 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 13179 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomeo 13180 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremoddprm 13181 A prime not equal to is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorempythagtriplem1 13182* Lemma for pythagtrip 13200. Prove a weaker version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem2 13183* Lemma for pythagtrip 13200. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem3 13184 Lemma for pythagtrip 13200. Show that and are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem4 13185 Lemma for pythagtrip 13200. Show that and are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem10 13186 Lemma for pythagtrip 13200. Show that is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem6 13187 Lemma for pythagtrip 13200. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem7 13188 Lemma for pythagtrip 13200. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem8 13189 Lemma for pythagtrip 13200. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem9 13190 Lemma for pythagtrip 13200. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem11 13191 Lemma for pythagtrip 13200. Show that (which will eventually be closely related to the in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem12 13192 Lemma for pythagtrip 13200. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem13 13193 Lemma for pythagtrip 13200. Show that (which will eventually be closely related to the in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem14 13194 Lemma for pythagtrip 13200. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem15 13195 Lemma for pythagtrip 13200. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem16 13196 Lemma for pythagtrip 13200. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem17 13197 Lemma for pythagtrip 13200. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem18 13198* Lemma for pythagtrip 13200. Wrap the previous and up in quanitifers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem19 13199* Lemma for pythagtrip 13200. Introduce and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtrip 13200* Parameterize the Pythagorean triples. If , , and are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. (Contributed by Scott Fenton, 19-Apr-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 31301-31400 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 32601-32689
 Copyright terms: Public domain < Previous  Next >