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Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
5.2.4  Properties of the canonical representation of a rational
 
Syntaxcnumer 12201 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 12202 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 12203* 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  =  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Definitiondf-denom 12204* 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  =  ( y  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumval 12205* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqdenval 12206* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumdencl 12207 Lemma for qnumcl 12208 and qdencl 12209. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 12208 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
 
Theoremqdencl 12209 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 12210 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 12211 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 12212 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.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( ( ( B 
 gcd  C )  =  1 
 /\  A  =  ( B  /  C ) )  <->  ( (numer `  A )  =  B  /\  (denom `  A )  =  C ) ) )
 
Theoremqnumdencoprm 12213 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  gcd  (denom `  A ) )  =  1
 )
 
Theoremqeqnumdivden 12214 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  A  =  ( (numer `  A )  /  (denom `  A ) ) )
 
Theoremqmuldeneqnum 12215 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A  x.  (denom `  A ) )  =  (numer `  A )
 )
 
Theoremdivnumden 12216 Calculate the reduced form of a quotient using  gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  ( A 
 /  ( A  gcd  B ) )  /\  (denom `  ( A  /  B ) )  =  ( B  /  ( A  gcd  B ) ) ) )
 
Theoremdivdenle 12217 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  (denom `  ( A  /  B ) )  <_  B )
 
Theoremqnumgt0 12218 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( 0  <  A  <->  0  <  (numer `  A ) ) )
 
Theoremqgt0numnn 12219 A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  0  <  A )  ->  (numer `  A )  e.  NN )
 
Theoremnn0gcdsq 12220 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
 
Theoremzgcdsq 12221 nn0gcdsq 12220 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 gcd  ( B ^
 2 ) ) )
 
Theoremnumdensq 12222 Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 )  /\  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) ) )
 
Theoremnumsq 12223 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 12224 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 12225 A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (denom `  A )  =  1  <->  A  e.  ZZ ) )
 
Theoremnn0sqrtelqelz 12226 If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.)
 |-  ( ( A  e.  NN0  /\  ( sqr `  A )  e.  QQ )  ->  ( sqr `  A )  e.  ZZ )
 
Theoremnonsq 12227 Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^ 2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2
 ) ) )  ->  -.  ( sqr `  A )  e.  QQ )
 
5.2.5  Euler's theorem
 
Syntaxcodz 12228 Extend class notation with the order function on the class of integers modulo N.
 class  odZ
 
Syntaxcphi 12229 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-odz 12230* Define the order function on the class of integers modulo N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
 |- 
 odZ  =  ( n  e.  NN  |->  ( x  e.  { x  e. 
 ZZ  |  ( x 
 gcd  n )  =  1 }  |-> inf ( { m  e.  NN  |  n  ||  ( ( x ^ m )  -  1
 ) } ,  RR ,  <  ) ) )
 
Definitiondf-phi 12231* Define the Euler phi function (also called "Euler totient function"), which counts the number of integers less than  n and coprime to it, see definition in [ApostolNT] p. 25. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 phi  =  ( n  e.  NN  |->  (♯ `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } ) )
 
Theoremphivalfi 12232* Finiteness of an expression used to define the Euler  phi function. (Contributed by Jim Kingon, 28-May-2022.)
 |-  ( N  e.  NN  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  e.  Fin )
 
Theoremphival 12233* Value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  =  (♯ `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } ) )
 
Theoremphicl2 12234 Bounds and closure for the value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  ( 1 ... N ) )
 
Theoremphicl 12235 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 12236* Lemma for phibnd 12237. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 12237 A slightly tighter bound on the value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( phi `  N )  <_  ( N  -  1
 ) )
 
Theoremphicld 12238 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( phi `  N )  e. 
 NN )
 
Theoremphi1 12239 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 12240* Alternate definition of the Euler 
phi function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( N  e.  NN  ->  ( phi `  N )  =  (♯ `  { x  e.  ( 0..^ N )  |  ( x  gcd  N )  =  1 } ) )
 
Theoremhashdvds 12241* 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.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ( ZZ>=
 `  ( A  -  1 ) ) )   &    |-  ( ph  ->  C  e.  ZZ )   =>    |-  ( ph  ->  (♯ ` 
 { x  e.  ( A ... B )  |  N  ||  ( x  -  C ) } )  =  ( ( |_ `  (
 ( B  -  C )  /  N ) )  -  ( |_ `  (
 ( ( A  -  1 )  -  C )  /  N ) ) ) )
 
Theoremphiprmpw 12242 Value of the Euler  phi function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^ ( K  -  1 ) )  x.  ( P  -  1
 ) ) )
 
Theoremphiprm 12243 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrth 12244* The Chinese Remainder Theorem: the function that maps  x to its remainder classes  mod  M and  mod  N is 1-1 and onto when  M and  N are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
 |-  S  =  ( 0..^ ( M  x.  N ) )   &    |-  T  =  ( ( 0..^ M )  X.  ( 0..^ N ) )   &    |-  F  =  ( x  e.  S  |->  <.
 ( x  mod  M ) ,  ( x  mod  N ) >. )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )
 )   =>    |-  ( ph  ->  F : S -1-1-onto-> T )
 
Theoremphimullem 12245* Lemma for phimul 12246. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  S  =  ( 0..^ ( M  x.  N ) )   &    |-  T  =  ( ( 0..^ M )  X.  ( 0..^ N ) )   &    |-  F  =  ( x  e.  S  |->  <.
 ( x  mod  M ) ,  ( x  mod  N ) >. )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )
 )   &    |-  U  =  { y  e.  ( 0..^ M )  |  ( y  gcd  M )  =  1 }   &    |-  V  =  { y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  W  =  { y  e.  S  |  ( y 
 gcd  ( M  x.  N ) )  =  1 }   =>    |-  ( ph  ->  ( phi `  ( M  x.  N ) )  =  ( ( phi `  M )  x.  ( phi `  N ) ) )
 
Theoremphimul 12246 The Euler  phi function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. Theorem 2.5(c) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( phi `  ( M  x.  N ) )  =  ( ( phi `  M )  x.  ( phi `  N ) ) )
 
Theoremeulerthlem1 12247* Lemma for eulerth 12253. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  T  =  ( 1 ... ( phi `  N ) )   &    |-  ( ph  ->  F : T
 -1-1-onto-> S )   &    |-  G  =  ( x  e.  T  |->  ( ( A  x.  ( F `  x ) ) 
 mod  N ) )   =>    |-  ( ph  ->  G : T --> S )
 
Theoremeulerthlemfi 12248* Lemma for eulerth 12253. The set  S is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   =>    |-  ( ph  ->  S  e.  Fin )
 
Theoremeulerthlemrprm 12249* Lemma for eulerth 12253. 
N and  prod_ x  e.  ( 1 ... ( phi `  N ) ) ( F `  x
) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  ( ph  ->  F : ( 1 ... ( phi `  N ) ) -1-1-onto-> S )   =>    |-  ( ph  ->  ( N  gcd  prod_ x  e.  (
 1 ... ( phi `  N ) ) ( F `
  x ) )  =  1 )
 
Theoremeulerthlema 12250* Lemma for eulerth 12253. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  ( ph  ->  F : ( 1 ... ( phi `  N ) ) -1-1-onto-> S )   =>    |-  ( ph  ->  ( (
 ( A ^ ( phi `  N ) )  x.  prod_ x  e.  (
 1 ... ( phi `  N ) ) ( F `
  x ) ) 
 mod  N )  =  (
 prod_ x  e.  (
 1 ... ( phi `  N ) ) ( ( A  x.  ( F `
  x ) ) 
 mod  N )  mod  N ) )
 
Theoremeulerthlemh 12251* Lemma for eulerth 12253. A permutation of  ( 1 ... ( phi `  N ) ). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  ( ph  ->  F : ( 1 ... ( phi `  N ) ) -1-1-onto-> S )   &    |-  H  =  ( `' F  o.  ( y  e.  ( 1 ... ( phi `  N ) ) 
 |->  ( ( A  x.  ( F `  y ) )  mod  N ) ) )   =>    |-  ( ph  ->  H : ( 1 ... ( phi `  N ) ) -1-1-onto-> ( 1 ... ( phi `  N ) ) )
 
Theoremeulerthlemth 12252* Lemma for eulerth 12253. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
 |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 ) )   &    |-  S  =  {
 y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }   &    |-  ( ph  ->  F : ( 1 ... ( phi `  N ) ) -1-1-onto-> S )   =>    |-  ( ph  ->  ( ( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod 
 N ) )
 
Theoremeulerth 12253 Euler's theorem, a generalization of Fermat's little theorem. If  A and  N are coprime, then  A ^ phi ( N )  ==  1 (mod  N). This is Metamath 100 proof #10. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( A ^
 ( phi `  N ) )  mod  N )  =  ( 1  mod 
 N ) )
 
Theoremfermltl 12254 Fermat's little theorem. When  P is prime,  A ^ P  ==  A (mod  P) for any  A, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( A ^ P )  mod  P )  =  ( A 
 mod  P ) )
 
Theoremprmdiv 12255 Show an explicit expression for the modular inverse of  A  mod  P. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  R  =  ( ( A ^ ( P  -  2 ) ) 
 mod  P )   =>    |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\ 
 -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  -  1 ) ) 
 /\  P  ||  (
 ( A  x.  R )  -  1 ) ) )
 
Theoremprmdiveq 12256 The modular inverse of  A  mod  P is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  R  =  ( ( A ^ ( P  -  2 ) ) 
 mod  P )   =>    |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\ 
 -.  P  ||  A )  ->  ( ( S  e.  ( 0 ... ( P  -  1
 ) )  /\  P  ||  ( ( A  x.  S )  -  1
 ) )  <->  S  =  R ) )
 
Theoremprmdivdiv 12257 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  R  =  ( ( A ^ ( P  -  2 ) ) 
 mod  P )   =>    |-  ( ( P  e.  Prime  /\  A  e.  (
 1 ... ( P  -  1 ) ) ) 
 ->  A  =  ( ( R ^ ( P  -  2 ) ) 
 mod  P ) )
 
Theoremhashgcdlem 12258* A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  A  =  { y  e.  ( 0..^ ( M 
 /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }   &    |-  B  =  { z  e.  ( 0..^ M )  |  ( z  gcd  M )  =  N }   &    |-  F  =  ( x  e.  A  |->  ( x  x.  N ) )   =>    |-  ( ( M  e.  NN  /\  N  e.  NN  /\  N  ||  M )  ->  F : A -1-1-onto-> B )
 
Theoremhashgcdeq 12259* Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  (♯ `  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
 )  =  if ( N  ||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) )
 
Theoremphisum 12260* The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  -> 
 sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( phi `  d )  =  N )
 
Theoremodzval 12261* Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod  N for some  N, 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  N. In order to ensure the supremum is well-defined, we only define the expression when  A and  N are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( odZ `  N ) `  A )  = inf ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1
 ) } ,  RR ,  <  ) )
 
Theoremodzcllem 12262 - Lemma for odzcl 12263, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( ( odZ `  N ) `  A )  e.  NN  /\  N  ||  ( ( A ^ ( ( odZ `  N ) `  A ) )  -  1 ) ) )
 
Theoremodzcl 12263 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( odZ `  N ) `  A )  e.  NN )
 
Theoremodzid 12264 Any element raised to the power of its order is  1. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  ||  ( ( A ^ ( ( odZ `  N ) `  A ) )  -  1 ) )
 
Theoremodzdvds 12265 The only powers of  A that are congruent to  1 are the multiples of the order of  A. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
 |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 )  /\  K  e.  NN0 )  ->  ( N  ||  ( ( A ^ K )  -  1
 ) 
 <->  ( ( odZ `  N ) `  A )  ||  K ) )
 
Theoremodzphi 12266 The order of any group element is a divisor of the Euler  phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( odZ `  N ) `  A )  ||  ( phi `  N ) )
 
5.2.6  Arithmetic modulo a prime number
 
Theoremmodprm1div 12267 A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( A 
 mod  P )  =  1  <->  P  ||  ( A  -  1 ) ) )
 
Theoremm1dvdsndvds 12268 If an integer minus 1 is divisible by a prime number, the integer itself is not divisible by this prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  -  1
 )  ->  -.  P  ||  A ) )
 
Theoremmodprminv 12269 Show an explicit expression for the modular inverse of  A  mod  P. This is an application of prmdiv 12255. (Contributed by Alexander van der Vekens, 15-May-2018.)
 |-  R  =  ( ( A ^ ( P  -  2 ) ) 
 mod  P )   =>    |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\ 
 -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  -  1 ) ) 
 /\  ( ( A  x.  R )  mod  P )  =  1 ) )
 
Theoremmodprminveq 12270 The modular inverse of  A  mod  P is unique. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  R  =  ( ( A ^ ( P  -  2 ) ) 
 mod  P )   =>    |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\ 
 -.  P  ||  A )  ->  ( ( S  e.  ( 0 ... ( P  -  1
 ) )  /\  (
 ( A  x.  S )  mod  P )  =  1 )  <->  S  =  R ) )
 
Theoremvfermltl 12271 Variant of Fermat's little theorem if  A is not a multiple of  P, see theorem 5.18 in [ApostolNT] p. 113. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 5-Sep-2020.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\ 
 -.  P  ||  A )  ->  ( ( A ^ ( P  -  1 ) )  mod  P )  =  1 )
 
Theorempowm2modprm 12272 If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  -  1
 )  ->  ( ( A ^ ( P  -  2 ) )  mod  P )  =  1 ) )
 
Theoremreumodprminv 12273* For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)
 |-  ( ( P  e.  Prime  /\  N  e.  (
 1..^ P ) ) 
 ->  E! i  e.  (
 1 ... ( P  -  1 ) ) ( ( N  x.  i
 )  mod  P )  =  1 )
 
Theoremmodprm0 12274* For two positive integers less than a given prime number there is always a nonnegative integer (less than the given prime number) so that the sum of one of the two positive integers and the other of the positive integers multiplied by the nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( P  e.  Prime  /\  N  e.  (
 1..^ P )  /\  I  e.  ( 1..^ P ) )  ->  E. j  e.  (
 0..^ P ) ( ( I  +  (
 j  x.  N ) )  mod  P )  =  0 )
 
Theoremnnnn0modprm0 12275* For a positive integer and a nonnegative integer both less than a given prime number there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the positive integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 8-Nov-2018.)
 |-  ( ( P  e.  Prime  /\  N  e.  (
 1..^ P )  /\  I  e.  ( 0..^ P ) )  ->  E. j  e.  (
 0..^ P ) ( ( I  +  (
 j  x.  N ) )  mod  P )  =  0 )
 
Theoremmodprmn0modprm0 12276* For an integer not being 0 modulo a given prime number and a nonnegative integer less than the prime number, there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 10-Nov-2018.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  ( N  mod  P )  =/=  0 )  ->  ( I  e.  (
 0..^ P )  ->  E. j  e.  (
 0..^ P ) ( ( I  +  (
 j  x.  N ) )  mod  P )  =  0 ) )
 
5.2.7  Pythagorean Triples
 
Theoremcoprimeprodsq 12277 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  gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A 
 gcd  B )  gcd  C )  =  1 )  ->  ( ( C ^
 2 )  =  ( A  x.  B ) 
 ->  A  =  ( ( A  gcd  C ) ^ 2 ) ) )
 
Theoremcoprimeprodsq2 12278 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  gcd and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B ) 
 gcd  C )  =  1 )  ->  ( ( C ^ 2 )  =  ( A  x.  B )  ->  B  =  ( ( B  gcd  C ) ^ 2 ) ) )
 
Theoremoddprm 12279 A prime not equal to  2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  ( ( N  -  1 )  / 
 2 )  e.  NN )
 
Theoremnnoddn2prm 12280 A prime not equal to  2 is an odd positive integer. (Contributed by AV, 28-Jun-2021.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  ( N  e.  NN  /\  -.  2  ||  N ) )
 
Theoremoddn2prm 12281 A prime not equal to  2 is odd. (Contributed by AV, 28-Jun-2021.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  -.  2  ||  N )
 
Theoremnnoddn2prmb 12282 A number is a prime number not equal to  2 iff it is an odd prime number. Conversion theorem for two representations of odd primes. (Contributed by AV, 14-Jul-2021.)
 |-  ( N  e.  ( Prime  \  { 2 } )  <->  ( N  e.  Prime  /\  -.  2  ||  N ) )
 
Theoremprm23lt5 12283 A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.)
 |-  ( ( P  e.  Prime  /\  P  <  5
 )  ->  ( P  =  2  \/  P  =  3 ) )
 
Theoremprm23ge5 12284 A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.)
 |-  ( P  e.  Prime  ->  ( P  =  2  \/  P  =  3  \/  P  e.  ( ZZ>= `  5 ) ) )
 
Theorempythagtriplem1 12285* Lemma for pythagtrip 12303. Prove a weaker version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( E. n  e. 
 NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  ( ( m ^
 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
 ( m ^ 2
 )  +  ( n ^ 2 ) ) ) )  ->  (
 ( A ^ 2
 )  +  ( B ^ 2 ) )  =  ( C ^
 2 ) )
 
Theorempythagtriplem2 12286* Lemma for pythagtrip 12303. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e. 
 NN  E. k  e.  NN  ( { A ,  B }  =  { (
 k  x.  ( ( m ^ 2 )  -  ( n ^
 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n ) ) ) }  /\  C  =  ( k  x.  (
 ( m ^ 2
 )  +  ( n ^ 2 ) ) ) )  ->  (
 ( A ^ 2
 )  +  ( B ^ 2 ) )  =  ( C ^
 2 ) ) )
 
Theorempythagtriplem3 12287 Lemma for pythagtrip 12303. Show that  C and 
B are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( B  gcd  C )  =  1 )
 
Theorempythagtriplem4 12288 Lemma for pythagtrip 12303. Show that  C  -  B and  C  +  B are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( ( C  -  B )  gcd  ( C  +  B ) )  =  1 )
 
Theorempythagtriplem10 12289 Lemma for pythagtrip 12303. Show that  C  -  B is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 ) )  ->  0  <  ( C  -  B ) )
 
Theorempythagtriplem6 12290 Lemma for pythagtrip 12303. Calculate  ( sqr `  ( C  -  B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( sqr `  ( C  -  B ) )  =  ( ( C  -  B )  gcd  A ) )
 
Theorempythagtriplem7 12291 Lemma for pythagtrip 12303. Calculate  ( sqr `  ( C  +  B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( sqr `  ( C  +  B ) )  =  ( ( C  +  B )  gcd  A ) )
 
Theorempythagtriplem8 12292 Lemma for pythagtrip 12303. Show that  ( sqr `  ( C  -  B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( sqr `  ( C  -  B ) )  e. 
 NN )
 
Theorempythagtriplem9 12293 Lemma for pythagtrip 12303. Show that  ( sqr `  ( C  +  B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( sqr `  ( C  +  B ) )  e. 
 NN )
 
Theorempythagtriplem11 12294 Lemma for pythagtrip 12303. Show that  M (which will eventually be closely related to the  m in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  M  =  ( ( ( sqr `  ( C  +  B )
 )  +  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  M  e.  NN )
 
Theorempythagtriplem12 12295 Lemma for pythagtrip 12303. Calculate the square of  M. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  M  =  ( ( ( sqr `  ( C  +  B )
 )  +  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( M ^ 2 )  =  ( ( C  +  A )  / 
 2 ) )
 
Theorempythagtriplem13 12296 Lemma for pythagtrip 12303. Show that  N (which will eventually be closely related to the  n in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  N  =  ( ( ( sqr `  ( C  +  B )
 )  -  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  N  e.  NN )
 
Theorempythagtriplem14 12297 Lemma for pythagtrip 12303. Calculate the square of  N. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  N  =  ( ( ( sqr `  ( C  +  B )
 )  -  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  ( N ^ 2 )  =  ( ( C  -  A )  / 
 2 ) )
 
Theorempythagtriplem15 12298 Lemma for pythagtrip 12303. Show the relationship between  M,  N, and  A. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  M  =  ( ( ( sqr `  ( C  +  B )
 )  +  ( sqr `  ( C  -  B ) ) )  / 
 2 )   &    |-  N  =  ( ( ( sqr `  ( C  +  B )
 )  -  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  A  =  ( ( M ^ 2 )  -  ( N ^ 2 ) ) )
 
Theorempythagtriplem16 12299 Lemma for pythagtrip 12303. Show the relationship between  M,  N, and  B. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  M  =  ( ( ( sqr `  ( C  +  B )
 )  +  ( sqr `  ( C  -  B ) ) )  / 
 2 )   &    |-  N  =  ( ( ( sqr `  ( C  +  B )
 )  -  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  B  =  ( 2  x.  ( M  x.  N ) ) )
 
Theorempythagtriplem17 12300 Lemma for pythagtrip 12303. Show the relationship between  M,  N, and  C. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  M  =  ( ( ( sqr `  ( C  +  B )
 )  +  ( sqr `  ( C  -  B ) ) )  / 
 2 )   &    |-  N  =  ( ( ( sqr `  ( C  +  B )
 )  -  ( sqr `  ( C  -  B ) ) )  / 
 2 )   =>    |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^ 2 )  +  ( B ^
 2 ) )  =  ( C ^ 2
 )  /\  ( ( A  gcd  B )  =  1  /\  -.  2  ||  A ) )  ->  C  =  ( ( M ^ 2 )  +  ( N ^ 2 ) ) )
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