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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
6.2.13  Ramsey's theorem
 
Syntaxcram 14601 Extend class notation with the Ramsey number function.
 class Ramsey
 
Theoremramtlecl 14602* The set  T of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  ph ) }   =>    |-  ( M  e.  T  ->  ( ZZ>= `  M )  C_  T )
 
Definitiondf-ram 14603* Define the Ramsey number function. The input is a number  m for the size of the edges of the hypergraph, and a tuple  r from the finite color set to lower bounds for each color. The Ramsey number  ( M Ramsey  R
) is the smallest number such that for any set  S with  ( M Ramsey  R
)  <_  # S and any coloring  F of the set of  M-element subsets of  S (with color set  dom  R), there is a color  c  e.  dom  R and a subset  x  C_  S such that  R ( c )  <_  # x and all the hyperedges of  x (that is, subsets of  x of size  M) have color  c. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |- Ramsey  =  ( m  e.  NN0 ,  r  e.  _V  |->  sup ( { n  e. 
 NN0  |  A. s ( n  <_  ( # `  s
 )  ->  A. f  e.  ( dom  r  ^m  { y  e.  ~P s  |  ( # `  y
 )  =  m }
 ) E. c  e. 
 dom  r E. x  e.  ~P  s ( ( r `  c ) 
 <_  ( # `  x )  /\  A. y  e. 
 ~P  x ( ( # `  y )  =  m  ->  ( f `  y )  =  c ) ) ) } ,  RR* ,  `'  <  ) )
 
Theoremhashbcval 14604* Value of the "binomial set", the set of all  N-element subsets of  A. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  V  /\  N  e.  NN0 )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x )  =  N }
 )
 
Theoremhashbccl 14605* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  Fin  /\  N  e.  NN0 )  ->  ( A C N )  e.  Fin )
 
Theoremhashbcss 14606* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  C_  ( A C N ) )
 
Theoremhashbc0 14607* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( A  e.  V  ->  ( A C 0 )  =  { (/) } )
 
Theoremhashbc2 14608* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  Fin  /\  N  e.  NN0 )  ->  ( # `  ( A C N ) )  =  ( ( # `  A )  _C  N ) )
 
Theorem0hashbc 14609* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( N  e.  NN  ->  ( (/) C N )  =  (/) )
 
Theoremramval 14610* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR*
 ,  `'  <  )
 )
 
Theoremramcl2lem 14611* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/)
 , +oo ,  sup ( T ,  RR ,  `'  <  ) ) )
 
Theoremramtcl 14612* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( ( M Ramsey  F )  e.  T  <->  T  =/=  (/) ) )
 
Theoremramtcl2 14613* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( ( M Ramsey  F )  e.  NN0  <->  T  =/=  (/) ) )
 
Theoremramtub 14614* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  A  e.  T )  ->  ( M Ramsey  F )  <_  A )
 
Theoremramub 14615* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  ( N  <_  ( # `
  s )  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e. 
 ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  N )
 
Theoremramub2 14616* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  ( ( # `  s
 )  =  N  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e. 
 ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  N )
 
Theoremrami 14617* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  ( M Ramsey  F )  e.  NN0 )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  ( M Ramsey  F )  <_  ( # `
  S ) )   &    |-  ( ph  ->  G :
 ( S C M )
 --> R )   =>    |-  ( ph  ->  E. c  e.  R  E. x  e. 
 ~P  S ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' G " { c } ) ) )
 
Theoremramcl2 14618 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  ( NN0  u.  { +oo } ) )
 
Theoremramxrcl 14619 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 14631.) (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
 
Theoremramubcl 14620 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( A  e.  NN0  /\  ( M Ramsey  F )  <_  A ) )  ->  ( M Ramsey  F )  e.  NN0 )
 
Theoremramlb 14621* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  G : ( ( 1
 ... N ) C M ) --> R )   &    |-  ( ( ph  /\  (
 c  e.  R  /\  x  C_  ( 1 ...
 N ) ) ) 
 ->  ( ( x C M )  C_  ( `' G " { c } )  ->  ( # `  x )  <  ( F `  c ) ) )   =>    |-  ( ph  ->  N  <  ( M Ramsey  F )
 )
 
Theorem0ram 14622* The Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( ( R  e.  V  /\  R  =/= 
 (/)  /\  F : R --> NN0 )  /\  E. x  e.  ZZ  A. y  e. 
 ran  F  y  <_  x )  ->  ( 0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
 
Theorem0ram2 14623 The Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R --> NN0 )  ->  (
 0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
 
Theoremram0 14624 The Ramsey number when  R  =  (/). (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( M  e.  NN0  ->  ( M Ramsey  (/) )  =  M )
 
Theorem0ramcl 14625 Lemma for ramcl 14631: Existence of the Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
 
Theoremramz2 14626 The Ramsey number when  F has value zero for some color  C. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
 --> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) ) 
 ->  ( M Ramsey  F )  =  0 )
 
Theoremramz 14627 The Ramsey number when  F is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
 0 } ) )  =  0 )
 
Theoremramub1lem1 14628* Lemma for ramub1 14630. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : R --> NN )   &    |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `  y
 ) ) ) ) )   &    |-  ( ph  ->  G : R --> NN0 )   &    |-  ( ph  ->  ( ( M  -  1 ) Ramsey  G )  e.  NN0 )   &    |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b )  =  i } )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( # `
  S )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )
 )   &    |-  ( ph  ->  K : ( S C M ) --> R )   &    |-  ( ph  ->  X  e.  S )   &    |-  H  =  ( u  e.  ( ( S  \  { X } ) C ( M  -  1 ) )  |->  ( K `  ( u  u.  { X } ) ) )   &    |-  ( ph  ->  D  e.  R )   &    |-  ( ph  ->  W 
 C_  ( S  \  { X } ) )   &    |-  ( ph  ->  ( G `  D )  <_  ( # `
  W ) )   &    |-  ( ph  ->  ( W C ( M  -  1 ) )  C_  ( `' H " { D } ) )   &    |-  ( ph  ->  E  e.  R )   &    |-  ( ph  ->  V  C_  W )   &    |-  ( ph  ->  if ( E  =  D ,  ( ( F `  D )  -  1
 ) ,  ( F `
  E ) ) 
 <_  ( # `  V ) )   &    |-  ( ph  ->  ( V C M ) 
 C_  ( `' K " { E } )
 )   =>    |-  ( ph  ->  E. z  e.  ~P  S ( ( F `  E ) 
 <_  ( # `  z
 )  /\  ( z C M )  C_  ( `' K " { E } ) ) )
 
Theoremramub1lem2 14629* Lemma for ramub1 14630. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : R --> NN )   &    |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `  y
 ) ) ) ) )   &    |-  ( ph  ->  G : R --> NN0 )   &    |-  ( ph  ->  ( ( M  -  1 ) Ramsey  G )  e.  NN0 )   &    |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b )  =  i } )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( # `
  S )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )
 )   &    |-  ( ph  ->  K : ( S C M ) --> R )   &    |-  ( ph  ->  X  e.  S )   &    |-  H  =  ( u  e.  ( ( S  \  { X } ) C ( M  -  1 ) )  |->  ( K `  ( u  u.  { X } ) ) )   =>    |-  ( ph  ->  E. c  e.  R  E. z  e. 
 ~P  S ( ( F `  c ) 
 <_  ( # `  z
 )  /\  ( z C M )  C_  ( `' K " { c } ) ) )
 
Theoremramub1 14630* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : R --> NN )   &    |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `  y
 ) ) ) ) )   &    |-  ( ph  ->  G : R --> NN0 )   &    |-  ( ph  ->  ( ( M  -  1 ) Ramsey  G )  e.  NN0 )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  (
 ( ( M  -  1 ) Ramsey  G )  +  1 ) )
 
Theoremramcl 14631 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  NN0 )
 
Theoremramsey 14632* Ramsey's theorem with the definition Ramsey eliminated. If  M is an integer,  R is a specified finite set of colors, and  F : R --> NN0 is a set of lower bounds for each color, then there is an  n such that for every set  s of size greater than  n and every coloring  f of the set  ( s C M ) of all  M-element subsets of  s, there is a color  c and a subset  x  C_  s such that  x is larger than  F (
c ) and the  M-element subsets of  x are monochromatic with color  c. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case  M  =  2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  E. n  e.  NN0  A. s ( n  <_  ( # `  s ) 
 ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) )
 
6.2.14  Decimal arithmetic (cont.)
 
Theoremdec2dvds 14633 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  ( B  x.  2
 )  =  C   &    |-  D  =  ( C  +  1 )   =>    |- 
 -.  2  || ; A D
 
Theoremdec5dvds 14634 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN   &    |-  B  <  5   =>    |- 
 -.  5  || ; A B
 
Theoremdec5dvds2 14635 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN   &    |-  B  <  5   &    |-  ( 5  +  B )  =  C   =>    |-  -.  5  || ; A C
 
Theoremdec5nprm 14636 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   =>    |-  -. ; A 5  e.  Prime
 
Theoremdec2nprm 14637 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  ( B  x.  2
 )  =  C   =>    |-  -. ; A C  e.  Prime
 
Theoremmodxai 14638 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  C  e.  NN0   &    |-  L  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( ( A ^ C )  mod  N )  =  ( L 
 mod  N )   &    |-  ( B  +  C )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  L )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmod2xi 14639 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( 2  x.  B )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  K )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmodxp1i 14640 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( B  +  1 )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  A )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmod2xnegi 14641 Version of mod2xi 14639 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN   &    |-  M  e.  NN0   &    |-  L  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( L  mod  N )   &    |-  ( 2  x.  B )  =  E   &    |-  ( L  +  K )  =  N   &    |-  ( ( D  x.  N )  +  M )  =  ( K  x.  K )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M 
 mod  N )
 
Theoremmodsubi 14642 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  mod  N )  =  ( K  mod  N )   &    |-  ( M  +  B )  =  K   =>    |-  (
 ( A  -  B )  mod  N )  =  ( M  mod  N )
 
Theoremgcdi 14643 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN0   &    |-  ( N  gcd  R )  =  G   &    |-  ( ( K  x.  N )  +  R )  =  M   =>    |-  ( M  gcd  N )  =  G
 
Theoremgcdmodi 14644 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN   &    |-  ( K  mod  N )  =  ( R  mod  N )   &    |-  ( N  gcd  R )  =  G   =>    |-  ( K  gcd  N )  =  G
 
Theoremdecexp2 14645 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  M  e.  NN0   &    |-  ( M  +  2 )  =  N   =>    |-  ( ( 4  x.  ( 2 ^ M ) )  +  0 )  =  (
 2 ^ N )
 
Theoremnumexp0 14646 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 14647 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
 
Theoremnumexpp1 14648 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A ^ M )  x.  A )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremnumexp2x 14649 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( 2  x.  M )  =  N   &    |-  ( A ^ M )  =  D   &    |-  ( D  x.  D )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremdecsplit0b 14650 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  B )  =  ( A  +  B )
 
Theoremdecsplit0 14651 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  0 )  =  A
 
Theoremdecsplit1 14652 Split a decimal number into two parts. Base case:  N  =  1. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 1 ) )  +  B )  = ; A B
 
Theoremdecsplit 14653 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  D  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A  x.  ( 10 ^ M ) )  +  B )  =  C   =>    |-  ( ( A  x.  ( 10 ^ N ) )  + ; B D )  = ; C D
 
Theoremkaratsuba 14654 The Karatsuba multiplication algorithm. If  X and 
Y are decomposed into two groups of digits of length  M (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then  X Y can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 11023. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  S  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  x.  C )  =  R   &    |-  ( B  x.  D )  =  T   &    |-  (
 ( A  +  B )  x.  ( C  +  D ) )  =  ( ( R  +  S )  +  T )   &    |-  ( ( A  x.  ( 10 ^ M ) )  +  B )  =  X   &    |-  ( ( C  x.  ( 10 ^ M ) )  +  D )  =  Y   &    |-  (
 ( R  x.  ( 10 ^ M ) )  +  S )  =  W   &    |-  ( ( W  x.  ( 10 ^ M ) )  +  T )  =  Z   =>    |-  ( X  x.  Y )  =  Z
 
Theorem2exp4 14655 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 4
 )  = ; 1 6
 
Theorem2exp6 14656 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp6OLD 14657 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of 2exp6 14656 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp8 14658 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 8
 )  = ;; 2 5 6
 
Theorem2exp16 14659 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
 
Theorem3exp3 14660 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 3 ^ 3
 )  = ; 2 7
 
Theorem2expltfac 14661 The factorial grows faster than two to the power  N. (Contributed by Mario Carneiro, 15-Sep-2016.)
 |-  ( N  e.  ( ZZ>=
 `  4 )  ->  ( 2 ^ N )  <  ( ! `  N ) )
 
6.2.15  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 14662 If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( ( L  e.  ZZ  /\  ( L  mod  ( # `  W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) )
 
Theoremcshwsidrepswmod0 14663 If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  (
 ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W ) )  =  0  \/  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) ) )
 
Theoremcshwshashlem1 14664* If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( W cyclShift  L )  =/=  W )
 
Theoremcshwshashlem2 14665* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  <  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwshashlem3 14666* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  =/=  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwsdisj 14667* The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 -> Disj 
 n  e.  ( 0..^ ( # `  W ) ) { ( W cyclShift  n ) } )
 
Theoremcshwsiun 14668* The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  =  U_ n  e.  ( 0..^ ( # `  W ) ) {
 ( W cyclShift  n ) }
 )
 
Theoremcshwsex 14669* The class of (different!) words resulting by cyclically shifting a given word is a set. (Contributed by AV, 8-Jun-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  e.  _V )
 
Theoremcshws0 14670* The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  =  (/)  ->  ( # `  M )  =  0 )
 
Theoremcshwrepswhash1 14671* The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( A  e.  V  /\  N  e.  NN  /\  W  =  ( A repeatS  N ) )  ->  ( # `  M )  =  1 )
 
Theoremcshwshashnsame 14672* If a word (not consisting of identical symbols) has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( E. i  e.  (
 0..^ ( # `  W ) ) ( W `
  i )  =/=  ( W `  0
 )  ->  ( # `  M )  =  ( # `  W ) ) )
 
Theoremcshwshash 14673* If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( ( # `  M )  =  ( # `  W )  \/  ( # `  M )  =  1 )
 )
 
6.2.16  Specific prime numbers
 
Theorem4nprm 14674 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  4  e.  Prime
 
Theoremprmlem0 14675* Lemma for prmlem1 14677 and prmlem2 14689. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( ( -.  2  ||  M  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2
 )  <_  N )  ->  -.  x  ||  N ) )   &    |-  ( K  e.  Prime  ->  -.  K  ||  N )   &    |-  ( K  +  2 )  =  M   =>    |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>= `  K ) )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )
 
Theoremprmlem1a 14676* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  (
 ( -.  2  ||  5  /\  x  e.  ( ZZ>=
 `  5 ) ) 
 ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )   =>    |-  N  e.  Prime
 
Theoremprmlem1 14677 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  N  < ; 2
 5   =>    |-  N  e.  Prime
 
Theorem5prm 14678 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  5  e.  Prime
 
Theorem6nprm 14679 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  6  e.  Prime
 
Theorem7prm 14680 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  7  e.  Prime
 
Theorem8nprm 14681 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  8  e.  Prime
 
Theorem9nprm 14682 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  9  e.  Prime
 
Theorem10nprm 14683 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  10  e.  Prime
 
Theorem11prm 14684 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 1  e.  Prime
 
Theorem13prm 14685 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 3  e.  Prime
 
Theorem17prm 14686 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 7  e.  Prime
 
Theorem19prm 14687 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 9  e.  Prime
 
Theorem23prm 14688 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 2
 3  e.  Prime
 
Theoremprmlem2 14689 Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than  5 ^ 2  =  2 5. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to  2 9 ^ 2  =  8 4 1, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 14702).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

 |-  N  e.  NN   &    |-  N  < ;; 8 4 1   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  -.  5  ||  N   &    |-  -.  7  ||  N   &    |- 
 -. ; 1 1  ||  N   &    |-  -. ; 1 3  ||  N   &    |-  -. ; 1 7 
 ||  N   &    |-  -. ; 1 9  ||  N   &    |-  -. ; 2 3 
 ||  N   =>    |-  N  e.  Prime
 
Theorem37prm 14690 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 3
 7  e.  Prime
 
Theorem43prm 14691 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 4
 3  e.  Prime
 
Theorem83prm 14692 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 8
 3  e.  Prime
 
Theorem139prm 14693 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 1 3 9  e. 
 Prime
 
Theorem163prm 14694 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 1 6 3  e. 
 Prime
 
Theorem317prm 14695 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 3 1 7  e. 
 Prime
 
Theorem631prm 14696 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 6 3 1  e. 
 Prime
 
6.2.17  Very large primes
 
Theorem1259lem1 14697 Lemma for 1259prm 14702. Calculate a power mod. In decimal, we calculate  2 ^ 1 6  =  5 2 N  +  6 8  ==  6 8 and  2 ^ 1 7  ==  6 8  x.  2  =  1 3 6 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 1 7 )  mod  N )  =  (;; 1 3 6  mod  N )
 
Theorem1259lem2 14698 Lemma for 1259prm 14702. Calculate a power mod. In decimal, we calculate  2 ^ 3 4  =  ( 2 ^ 1 7 ) ^ 2  ==  1
3 6 ^ 2  ==  1 4 N  +  8 7 0. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 3 4 )  mod  N )  =  (;; 8 7 0  mod  N )
 
Theorem1259lem3 14699 Lemma for 1259prm 14702. Calculate a power mod. In decimal, we calculate  2 ^ 3 8  =  2 ^ 3 4  x.  2 ^ 4  ==  8
7 0  x.  1 6  =  1 1 N  +  7 1 and  2 ^ 7 6  =  ( 2 ^ 3 4 ) ^ 2  ==  7
1 ^ 2  =  4 N  +  5  ==  5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 7 6 )  mod  N )  =  ( 5  mod 
 N )
 
Theorem1259lem4 14700 Lemma for 1259prm 14702. Calculate a power mod. In decimal, we calculate  2 ^ 3 0 6  =  ( 2 ^ 7 6 ) ^ 4  x.  4  ==  5 ^ 4  x.  4  =  2 N  -  1 8,  2 ^ 6 1 2  =  ( 2 ^ 3 0 6 ) ^ 2  ==  1 8 ^ 2  =  3 2 4,  2 ^ 6 2 9  =  2 ^ 6 1 2  x.  2 ^ 1 7  ==  3 2 4  x.  1 3 6  =  3 5 N  -  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 6 2 9 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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