HomeHome Metamath Proof Explorer
Theorem List (p. 129 of 322)
< 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  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpexp12i 12801 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 )
 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )
 
Theoremrpmul 12802 If  K is relatively prime to  M and to  N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K 
 gcd  M )  =  1 
 /\  ( K  gcd  N )  =  1 ) 
 ->  ( K  gcd  ( M  x.  N ) )  =  1 ) )
 
Theoremrpdvds 12803 If  K is relatively prime to  N then it is also relatively prime to any divisor  M of  N. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
 
6.2.2  Properties of the canonical representation of a rational
 
Syntaxcnumer 12804 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 12805 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 12806* 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 12807* 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 12808* 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 12809* 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 12810 Lemma for qnumcl 12811 and qdencl 12812. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 12811 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 12812 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 12813 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 12814 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 12815 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 12816 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 12817 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 12818 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 12819 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 12820 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 12821 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 12822 A rational is positive iff its canonical numerator is a natural number. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  0  <  A )  ->  (numer `  A )  e.  NN )
 
Theoremnn0gcdsq 12823 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 12824 nn0gcdsq 12823 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 12825 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 12826 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 12827 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 12828 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 ) )
 
Theoremzsqrelqelz 12829 If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( sqr `  A )  e.  QQ )  ->  ( sqr `  A )  e.  ZZ )
 
Theoremnonsq 12830 Any integer strictly between two adjacent squares has an irrational 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 )
 
6.2.3  Euler's theorem
 
Syntaxcodz 12831 Extend class notation with the order function on the class of integers mod N.
 class  od Z
 
Syntaxcphi 12832 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-odz 12833* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 od Z  =  ( n  e.  NN  |->  ( x  e.  { x  e. 
 ZZ  |  ( x 
 gcd  n )  =  1 }  |->  sup ( { m  e.  NN  |  n  ||  ( ( x ^ m )  -  1 ) } ,  RR ,  `'  <  ) ) )
 
Definitiondf-phi 12834* Define the Euler phi function, which counts the number of integers less than  n and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 phi  =  ( n  e.  NN  |->  ( # `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } ) )
 
Theoremphival 12835* 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 12836 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 12837 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 12838* Lemma for phibnd 12839. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 12839 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 12840 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 12841 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 12842* 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 12843* 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 12844 Value of the Euler  phi function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^ ( K  -  1 ) )  x.  ( P  -  1
 ) ) )
 
Theoremphiprm 12845 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrt 12846* 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 12847* Lemma for phimul 12848. (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 12848 The Euler  phi function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. (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 12849* Lemma for eulerth 12851. (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 )
 
Theoremeulerthlem2 12850* Lemma for eulerth 12851. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( 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  ->  ( ( A ^ ( phi `  N ) ) 
 mod  N )  =  ( 1  mod  N ) )
 
Theoremeulerth 12851 Euler's theorem, a generalization of Fermat's little theorem. If  A and  N are coprime, then  A ^ phi ( N )  ==  1, mod  N. (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 12852 Fermat's little theorem. When  P is prime,  A ^ P  ==  A, mod  P for any  A. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( A ^ P )  mod  P )  =  ( A 
 mod  P ) )
 
Theoremprmdiv 12853 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 12854 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 12855 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 ) )
 
Theoremodzval 12856* 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.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( od Z `  N ) `  A )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1
 ) } ,  RR ,  `'  <  ) )
 
Theoremodzcllem 12857 - Lemma for odzcl 12858, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( ( od
 Z `  N ) `  A )  e.  NN  /\  N  ||  ( ( A ^ ( ( od
 Z `  N ) `  A ) )  -  1 ) ) )
 
Theoremodzcl 12858 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 )  ->  ( ( od Z `  N ) `  A )  e.  NN )
 
Theoremodzid 12859 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 ^ ( ( od
 Z `  N ) `  A ) )  -  1 ) )
 
Theoremodzdvds 12860 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.)
 |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 )  /\  K  e.  NN0 )  ->  ( N  ||  ( ( A ^ K )  -  1
 ) 
 <->  ( ( od Z `  N ) `  A )  ||  K ) )
 
Theoremodzphi 12861 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 )  ->  ( ( od Z `  N ) `  A )  ||  ( phi `  N ) )
 
6.2.4  Pythagorean Triples
 
Theoremcoprimeprodsq 12862 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 12863 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 ) ) )
 
Theoremopoe 12864 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  +  B ) )
 
Theoremomoe 12865 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  -  B ) )
 
Theoremopeo 12866 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\  2  ||  B )
 )  ->  -.  2  ||  ( A  +  B ) )
 
Theoremomeo 12867 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\  2  ||  B )
 )  ->  -.  2  ||  ( A  -  B ) )
 
Theoremoddprm 12868 A prime not equal to  2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  ( ( N  -  1 )  / 
 2 )  e.  NN )
 
Theorempythagtriplem1 12869* Lemma for pythagtrip 12887. 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 12870* Lemma for pythagtrip 12887. 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 12871 Lemma for pythagtrip 12887. 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 12872 Lemma for pythagtrip 12887. 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 12873 Lemma for pythagtrip 12887. 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 12874 Lemma for pythagtrip 12887. 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 12875 Lemma for pythagtrip 12887. 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 12876 Lemma for pythagtrip 12887. Show that  ( sqr `  ( C  -  B ) ) is a natural number (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 12877 Lemma for pythagtrip 12887. Show that  ( sqr `  ( C  +  B ) ) is a natural number (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 12878 Lemma for pythagtrip 12887. 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 12879 Lemma for pythagtrip 12887. 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 12880 Lemma for pythagtrip 12887. 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 12881 Lemma for pythagtrip 12887. 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 12882 Lemma for pythagtrip 12887. 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 12883 Lemma for pythagtrip 12887. 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 12884 Lemma for pythagtrip 12887. 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 ) ) )
 
Theorempythagtriplem18 12885* Lemma for pythagtrip 12887. Wrap the previous  M and  N up in quanitifers. (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 ) )  ->  E. n  e.  NN  E. m  e.  NN  ( A  =  ( ( m ^ 2 )  -  ( n ^ 2 ) )  /\  B  =  ( 2  x.  ( m  x.  n ) ) 
 /\  C  =  ( ( m ^ 2
 )  +  ( n ^ 2 ) ) ) )
 
Theorempythagtriplem19 12886* Lemma for pythagtrip 12887. Introduce  k and remove the relative primality requirement. (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
 )  /\  -.  2  ||  ( A  /  ( A  gcd  B ) ) )  ->  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 ) ) ) ) )
 
Theorempythagtrip 12887* Parameterize the Pythagorean triples. If  A,  B, and  C 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.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  <->  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 ) ) ) ) ) )
 
Theoremiserodd 12888* Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( ph  /\  k  e.  NN0 )  ->  C  e.  CC )   &    |-  ( n  =  ( ( 2  x.  k )  +  1 )  ->  B  =  C )   =>    |-  ( ph  ->  (  seq  0 (  +  ,  ( k  e.  NN0  |->  C ) )  ~~>  A  <->  seq  1 (  +  ,  ( n  e.  NN  |->  if ( 2  ||  n ,  0 ,  B ) ) )  ~~>  A )
 )
 
6.2.5  The prime count function
 
Syntaxcpc 12889 Extend class notation with the prime count function.
 class  pCnt
 
Definitiondf-pc 12890* Define the prime count function, which returns the largest exponent of a given prime (or other natural number) that divides the number. For rational numbers, it returns negative values according to the power of a prime in the denominator. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 pCnt  =  ( p  e.  Prime ,  r  e. 
 QQ  |->  if ( r  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e. 
 NN  ( r  =  ( x  /  y
 )  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n ) 
 ||  y } ,  RR ,  <  ) ) ) ) ) )
 
Theorempclem 12891* - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( A  C_ 
 ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y 
 <_  x ) )
 
Theorempcprecl 12892* Closure of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   &    |-  S  =  sup ( A ,  RR ,  <  )   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( S  e.  NN0  /\  ( P ^ S )  ||  N ) )
 
Theorempcprendvds 12893* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   &    |-  S  =  sup ( A ,  RR ,  <  )   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ ( S  +  1 ) )  ||  N )
 
Theorempcprendvds2 12894* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   &    |-  S  =  sup ( A ,  RR ,  <  )   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ S ) ) )
 
Theorempcpre1 12895* Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   &    |-  S  =  sup ( A ,  RR ,  <  )   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  =  1 ) 
 ->  S  =  0 )
 
Theorempcpremul 12896* Multiplicative property of the prime count pre-function. Note that the primality of  P is essential for this property;  ( 4  pCnt  2
)  =  0 but  ( 4  pCnt 
( 2  x.  2 ) )  =  1  =/=  2  x.  (
4  pCnt  2 )  =  0. Since this is needed to show uniqueness for the real prime count function (over  QQ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n ) 
 ||  M } ,  RR ,  <  )   &    |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  )   &    |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N ) } ,  RR ,  <  )   =>    |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( S  +  T )  =  U )
 
Theorempcval 12897* The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
 |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n ) 
 ||  x } ,  RR ,  <  )   &    |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )   =>    |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0
 ) )  ->  ( P  pCnt  N )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
 )  /\  z  =  ( S  -  T ) ) ) )
 
Theorempceulem 12898* Lemma for pceu 12899. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n ) 
 ||  x } ,  RR ,  <  )   &    |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )   &    |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )   &    |-  V  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  =/=  0 )   &    |-  ( ph  ->  ( x  e. 
 ZZ  /\  y  e.  NN ) )   &    |-  ( ph  ->  N  =  ( x  /  y ) )   &    |-  ( ph  ->  ( s  e. 
 ZZ  /\  t  e.  NN ) )   &    |-  ( ph  ->  N  =  ( s  /  t ) )   =>    |-  ( ph  ->  ( S  -  T )  =  ( U  -  V ) )
 
Theorempceu 12899* Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n ) 
 ||  x } ,  RR ,  <  )   &    |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )   =>    |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0
 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
 
Theorempczpre 12900* Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n ) 
 ||  N } ,  RR ,  <  )   =>    |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( P  pCnt  N )  =  S )
    < Previous  Next >

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 32101-32154
  Copyright terms: Public domain < Previous  Next >