arara:# gp
        GP/PARI CALCULATOR Version 2.12.1 (development 25142-46eb34041)
          i386 running darwin (x86-64/GMP-6.2.0 kernel) 64-bit version
     compiled: Mar 22 2020, Apple clang version 11.0.0 (clang-1100.0.33.17)
                            threading engine: single
                 (readline v8.0 enabled, extended help enabled)

                     Copyright (C) 2000-2020 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes 
WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000
? ?Mod
Mod(a,b): create 'a modulo b'.

? ?lift
lift(x,{v}): if v is omitted, lifts elements of Z/nZ to Z, of Qp to Q, and of 
K[x]/(P) to K[x]. Otherwise lift only polmods with main variable v.

? ?liftint
liftint(x): lifts every element of Z/nZ to Z and of Qp to Q.

? Mod(2*2,3)
%1 = Mod(1, 3)
? liftint(Mod(2*2,3))
%2 = 1
? p=3
%3 = 3
? matrix(p,p,a,b,a+b)
%4 = 
[2 3 4]

[3 4 5]

[4 5 6]

? matrix(p,p,a,b,Mod(a+b,p))
%5 = 
[Mod(2, 3) Mod(0, 3) Mod(1, 3)]

[Mod(0, 3) Mod(1, 3) Mod(2, 3)]

[Mod(1, 3) Mod(2, 3) Mod(0, 3)]

? matrix(p,p,a,b,liftint(Mod(a+b,p)))
%6 = 
[2 0 1]

[0 1 2]

[1 2 0]

? matrix(p,p,a,b,s(a+b))
  ***   at top-level: matrix(p,p,a,b,s(a+b))
  ***                                ^-------
  ***   not a function in function call
  ***   Break loop: type 'break' to go back to GP prompt
break> break

? s(x) = liftint(Mod(x,p))
%7 = (x)->liftint(Mod(x,p))
? matrix(p,p,a,b,s(a+b))
%8 = 
[2 0 1]

[0 1 2]

[1 2 0]

? p=4;matrix(p,p,a,b,s(a+b))
%9 = 
[2 3 0 1]

[3 0 1 2]

[0 1 2 3]

[1 2 3 0]

? p=5;matrix(p,p,a,b,s(a+b))
%10 = 
[2 3 4 0 1]

[3 4 0 1 2]

[4 0 1 2 3]

[0 1 2 3 4]

[1 2 3 4 0]

? p=6;matrix(p,p,a,b,s(a+b))
%11 = 
[2 3 4 5 0 1]

[3 4 5 0 1 2]

[4 5 0 1 2 3]

[5 0 1 2 3 4]

[0 1 2 3 4 5]

[1 2 3 4 5 0]

? p=7;matrix(p,p,a,b,s(a+b))
%12 = 
[2 3 4 5 6 0 1]

[3 4 5 6 0 1 2]

[4 5 6 0 1 2 3]

[5 6 0 1 2 3 4]

[6 0 1 2 3 4 5]

[0 1 2 3 4 5 6]

[1 2 3 4 5 6 0]

? p=8;matrix(p,p,a,b,s(a+b))
%13 = 
[2 3 4 5 6 7 0 1]

[3 4 5 6 7 0 1 2]

[4 5 6 7 0 1 2 3]

[5 6 7 0 1 2 3 4]

[6 7 0 1 2 3 4 5]

[7 0 1 2 3 4 5 6]

[0 1 2 3 4 5 6 7]

[1 2 3 4 5 6 7 0]

? p=2;matrix(p,p,a,b,s(a*b))
%14 = 
[1 0]

[0 0]

? p=3;matrix(p,p,a,b,s(a*b))
%15 = 
[1 2 0]

[2 1 0]

[0 0 0]

? M(p) = matrix(p,p,a,b,s(a*b))
%16 = (p)->matrix(p,p,a,b,s(a*b))
? M(2)
%17 = 
[1 2]

[2 1]

? ?s
s =
  (x)->liftint(Mod(x,p))

? kill(p)
? M(p) = matrix(p,p,a,b,s(a*b))
%19 = (p)->matrix(p,p,a,b,s(a*b))
? M(2)
%20 = 
[Mod(1, p) Mod(2, p)]

[Mod(2, p) Mod(4, p)]

? ?s
s =
  (x)->liftint(Mod(x,p))

? s(x,p) = liftint(Mod(x,p))
%21 = (x,p)->liftint(Mod(x,p))
? M(p) = matrix(p,p,a,b,s(a*b,p))
%22 = (p)->matrix(p,p,a,b,s(a*b,p))
? M(2)
%23 = 
[1 0]

[0 0]

? ?M
M =
  (p)->matrix(p,p,a,b,s(a*b,p))

? ?s
s =
  (x,p)->liftint(Mod(x,p))

? "s para representante Mod p enter 0 e p-1"
%24 = "s para representante Mod p enter 0 e p-1"
? "M(p) e uma tabua de multiplicação mod p para um inteiro p"
%25 = "M(p) e uma tabua de multiplicação mod p para um inteiro p"
? Mod(4,-5)
%26 = Mod(4, 5)
? Mod(-4,-5)
%27 = Mod(1, 5)
? s(-2)
  ***   at top-level: s(-2)
  ***                 ^-----
  ***   in function s: liftint(Mod(x,p))
  ***                          ^---------
  *** Mod: impossible inverse in %: 0.
  ***   Break loop: type 'break' to go back to GP prompt
break> break

? s(2)
  ***   at top-level: s(2)
  ***                 ^----
  ***   in function s: liftint(Mod(x,p))
  ***                          ^---------
  *** Mod: impossible inverse in %: 0.
  ***   Break loop: type 'break' to go back to GP prompt
break> break

? M(-2)
  ***   at top-level: M(-2)
  ***                 ^-----
  ***   in function M: matrix(p,p,a,b,s(a*b,p))
  ***                  ^------------------------
  *** matrix: domain error in matrix: nbcols < 0
  ***   Break loop: type 'break' to go back to GP prompt
break> break

? M(2)
%28 = 
[1 0]

[0 0]

? ?M
M =
  (p)->matrix(p,p,a,b,s(a*b,p))

? Mod(4,5) == Mod(4,-5)
%29 = 1
? M(p) = matrix(abs(p),abs(p),a,b,s(a*b,p))
%30 = (p)->matrix(abs(p),abs(p),a,b,s(a*b,p))
? ?abs
abs(x): absolute value (or modulus) of x.

? abs(-5)
%31 = 5
? M(-2)
%32 = 
[1 0]

[0 0]

? M(-2) == M(2)
%33 = 1
? M(3)
%34 = 
[1 2 0]

[2 1 0]

[0 0 0]

? ?M
M =
  (p)->matrix(abs(p),abs(p),a,b,s(a*b,p))

? ?matrix
matrix(m,{n=m},{X},{Y},{expr=0}): m x n matrix of expression expr, where the 
row variable X goes from 1 to m and the column variable Y goes from 1 to n. By 
default, fill with 0s.

? "variavel de reta vai 1 ate p, então 0 aqui é no fim de tabua"
%35 = "variavel de reta vai 1 ate p, então 0 aqui é no fim de tabua"
? ?mat
matadjoint        mathess           matintersect      matqr
matalgtobasis     mathilbert        matinverseimage   matrank
matbasistoalg     mathnf            matinvmod         matreduce
matcompanion      mathnfmod         matisdiagonal     matrix
matconcat         mathnfmodid       matker            matrixqz
matdet            mathouseholder    matkerint         matsize
matdetint         matid             matkermod         matsnf
matdetmod         matimage          matmuldiagonal    matsolve
matdiagonal       matimagecompl     matmultodiagonal  matsolvemod
mateigen          matimagemod       matpascal         matsupplement
matfrobenius      matindexrank      matpermanent      mattranspose
? ?M
M =
  (p)->matrix(abs(p),abs(p),a,b,s(a*b,p))

? N(p) = matrix(abs(p)-1,abs(p)-1,a,b,s(a*b,p))
%36 = (p)->matrix(abs(p)-1,abs(p)-1,a,b,s(a*b,p))
? N(3)
%37 = 
[1 2]

[2 1]

? M(3)
%38 = 
[1 2 0]

[2 1 0]

[0 0 0]

? MM(p) = matrix(abs(p),abs(p),a,b,s((a-1)*(b-1),p))
%39 = (p)->matrix(abs(p),abs(p),a,b,s((a-1)*(b-1),p))
? M(3)
%40 = 
[1 2 0]

[2 1 0]

[0 0 0]

? MM(3)
%41 = 
[0 0 0]

[0 1 2]

[0 2 1]

? N(2)
%42 = 
[1]

? N(3)
%43 = 
[1 2]

[2 1]

? N(4)
%44 = 
[1 2 3]

[2 0 2]

[3 2 1]

? M(4)
%45 = 
[1 2 3 0]

[2 0 2 0]

[3 2 1 0]

[0 0 0 0]

? MM(4)
%46 = 
[0 0 0 0]

[0 1 2 3]

[0 2 0 2]

[0 3 2 1]

? ?mat
matadjoint        mathess           matintersect      matqr
matalgtobasis     mathilbert        matinverseimage   matrank
matbasistoalg     mathnf            matinvmod         matreduce
matcompanion      mathnfmod         matisdiagonal     matrix
matconcat         mathnfmodid       matker            matrixqz
matdet            mathouseholder    matkerint         matsize
matdetint         matid             matkermod         matsnf
matdetmod         matimage          matmuldiagonal    matsolve
matdiagonal       matimagecompl     matmultodiagonal  matsolvemod
mateigen          matimagemod       matpascal         matsupplement
matfrobenius      matindexrank      matpermanent      mattranspose
? ?vec
vecextract   vecmin       vecsearch    vecsum       vectorsmall  
vecmax       vecprod      vecsort      vector       vectorv      
? ?vecextract 
vecextract(x,y,{z}): extraction of the components of the matrix or vector x 
according to y and z. If z is omitted, y represents columns, otherwise y 
corresponds to rows and z to columns. y and z can be vectors (of indices), 
strings (indicating ranges as in "1..10") or masks (integers whose binary 
representation indicates the indices to extract, from left to right 1, 2, 4, 8, 
etc.).

? MM(4)
%47 = 
[0 0 0 0]

[0 1 2 3]

[0 2 0 2]

[0 3 2 1]

? %
%48 = 
[0 0 0 0]

[0 1 2 3]

[0 2 0 2]

[0 3 2 1]

? %48 == %47
%49 = 1
? %47
%50 = 
[0 0 0 0]

[0 1 2 3]

[0 2 0 2]

[0 3 2 1]

? vecextract(%,[2,4],[2,4])
%51 = 
[1 3]

[3 1]

? M(4)
%52 = 
[1 2 3 0]

[2 0 2 0]

[3 2 1 0]

[0 0 0 0]

? vecextract(M(4),[1,3],[1,3])
%53 = 
[1 3]

[3 1]

? "Modulo 4 residuous intersiveis são 1 e 3, e eles formam um grupo de 2 elemen
tos, o mesmo grupo que a gente já viu como residuous modulo 2 com + e tb residu
os modulo inversiveis com *"
%54 = "Modulo 4 residuous intersiveis são 1 e 3, e eles formam um grupo de 2 elementos, o mesmo grupo que a gente já viu como residuous modulo 2 com + e tb residuos modulo inversiveis com *"
? M(5)
%55 = 
[1 2 3 4 0]

[2 4 1 3 0]

[3 1 4 2 0]

[4 3 2 1 0]

[0 0 0 0 0]

? vecextract(M(5),"1..4",[1,2,3,4])
%56 = 
[1 2 3 4]

[2 4 1 3]

[3 1 4 2]

[4 3 2 1]

? ?vecextract
vecextract(x,y,{z}): extraction of the components of the matrix or vector x 
according to y and z. If z is omitted, y represents columns, otherwise y 
corresponds to rows and z to columns. y and z can be vectors (of indices), 
strings (indicating ranges as in "1..10") or masks (integers whose binary 
representation indicates the indices to extract, from left to right 1, 2, 4, 8, 
etc.).

? A(n) = matrix(n,n,a,b,liftint(Mod((a-1)+(b-1),n)))
%57 = (n)->matrix(n,n,a,b,liftint(Mod((a-1)+(b-1),n)))
? A(4)
%58 = 
[0 1 2 3]

[1 2 3 0]

[2 3 0 1]

[3 0 1 2]

? N(5)
%59 = 
[1 2 3 4]

[2 4 1 3]

[3 1 4 2]

[4 3 2 1]

? "Aqui uma pergunta - estes 2 tabuas podem ser iguas depois renomeamento?"
%60 = "Aqui uma pergunta - estes 2 tabuas podem ser iguas depois renomeamento?"
? Letras = [a,b,c,d]
%61 = [a, b, c, d]
? N(5)
%62 = 
[1 2 3 4]

[2 4 1 3]

[3 1 4 2]

[4 3 2 1]

? matrix(4,4,u,v,N(5)[u,v])
%63 = 
[1 2 3 4]

[2 4 1 3]

[3 1 4 2]

[4 3 2 1]

? matrix(4,4,u,v,Letras[N(5)[u,v]])
%64 = 
[a b c d]

[b d a c]

[c a d b]

[d c b a]

? N(5)
%65 = 
[1 2 3 4]

[2 4 1 3]

[3 1 4 2]

[4 3 2 1]

? M(8)
%66 = 
[1 2 3 4 5 6 7 0]

[2 4 6 0 2 4 6 0]

[3 6 1 4 7 2 5 0]

[4 0 4 0 4 0 4 0]

[5 2 7 4 1 6 3 0]

[6 4 2 0 6 4 2 0]

[7 6 5 4 3 2 1 0]

[0 0 0 0 0 0 0 0]

? vecextract(M(8),[1,3,5,7],[1,3,5,7])
%67 = 
[1 3 5 7]

[3 1 7 5]

[5 7 1 3]

[7 5 3 1]

? "Quais destes grupos de 4 elementos são isomorfos um doutro: Z/4, (Z/5)^*, (Z
/8)^* ?"
%68 = "Quais destes grupos de 4 elementos são isomorfos um doutro: Z/4, (Z/5)^*, (Z/8)^* ?"
? matrix(4,4,u,v,Letras[N(5)[u,v]])
%69 = 
[a b c d]

[b d a c]

[c a d b]

[d c b a]

? M5literal = matrix(4,4,u,v,Letras[N(5)[u,v]])
%70 = 
[a b c d]

[b d a c]

[c a d b]

[d c b a]

? subtvec(M5literal, [a,b,c,d], [Sergey, Rafael, Felipe, Tomás])
  ***   syntax error, unexpected $undefined, expecting ',': 
  ***   ...rgey,Rafael,Felipe,Tomás])
  ***                               ^--
? subtvec(M5literal, [a,b,c,d], [Sergey, Rafael, Felipe, Tomas])
  ***   at top-level: subtvec(M5literal,[a,b,c,d],[Sergey,Rafael,Fel
  ***                 ^----------------------------------------------
  ***   not a function in function call
  ***   Break loop: type 'break' to go back to GP prompt
break> break

? substvec(M5literal, [a,b,c,d], [Sergey, Rafael, Felipe, Tomás])
  ***   syntax error, unexpected $undefined, expecting ',': 
  ***   ...rgey,Rafael,Felipe,Tomás])
  ***                               ^--
? substvec(M5literal, [a,b,c,d], [Sergey, Rafael, Felipe, Tomas])
%71 = 
[Sergey Rafael Felipe  Tomas]

[Rafael  Tomas Sergey Felipe]

[Felipe Sergey  Tomas Rafael]

[ Tomas Felipe Rafael Sergey]

? subst(%,Tomas,Giovanna)
%72 = 
[  Sergey   Rafael   Felipe Giovanna]

[  Rafael Giovanna   Sergey   Felipe]

[  Felipe   Sergey Giovanna   Rafael]

[Giovanna   Felipe   Rafael   Sergey]

? subst(%,Rafael,Raphael)
%73 = 
[  Sergey  Raphael   Felipe Giovanna]

[ Raphael Giovanna   Sergey   Felipe]

[  Felipe   Sergey Giovanna  Raphael]

[Giovanna   Felipe  Raphael   Sergey]

? subst(%,Sergey, Rafael)
%74 = 
[  Rafael  Raphael   Felipe Giovanna]

[ Raphael Giovanna   Rafael   Felipe]

[  Felipe   Rafael Giovanna  Raphael]

[Giovanna   Felipe  Raphael   Rafael]

? "A gente pode usar subst e substvec para renomear elementos, mas para mudar o
rdem deles a gente precisa outra operação"
%75 = "A gente pode usar subst e substvec para renomear elementos, mas para mudar ordem deles a gente precisa outra operação"
? S12 = [0,1,0,0;1,0,0,0;0,0,1,0;0,0,0,1]
%76 = 
[0 1 0 0]

[1 0 0 0]

[0 0 1 0]

[0 0 0 1]

? S12 * %74 * S12
%77 = 
[Giovanna  Raphael   Rafael   Felipe]

[ Raphael   Rafael   Felipe Giovanna]

[  Rafael   Felipe Giovanna  Raphael]

[  Felipe Giovanna  Raphael   Rafael]

? %74
%78 = 
[  Rafael  Raphael   Felipe Giovanna]

[ Raphael Giovanna   Rafael   Felipe]

[  Felipe   Rafael Giovanna  Raphael]

[Giovanna   Felipe  Raphael   Rafael]

? vector(8,n,Mod(n^2,8))
%79 = [Mod(1, 8), Mod(4, 8), Mod(1, 8), Mod(0, 8), Mod(1, 8), Mod(4, 8), Mod(1, 8), Mod(0, 8)]
? liftint(%)
%80 = [1, 4, 1, 0, 1, 4, 1, 0]
? "NB: Para cado número impar seu quadrado igual 1 modulo 8"
%81 = "NB: Para cado número impar seu quadrado igual 1 modulo 8"
? Mod(2+2,4)
%82 = Mod(0, 4)
? Mod(1+1,4)
%83 = Mod(2, 4)
? a = Mod(1,4)
%84 = Mod(1, 4)
? a+a
%85 = Mod(2, 4)
? a+a+a
%86 = Mod(3, 4)
? a+a+a+a
%87 = Mod(0, 4)
? 
Goodbye!