#OK to Post

#1: Sample of Examples from pages 30-60 of Garvan's booklet

#Data Types: 

#SEQUENCE: Finding nth roots of unity
s:=seq(x^i-1, i=1..5);
for i from 1 to 5 do factor(s[i])
end do;
for i from 1 to 5 do fsolve(s[i],x, complex)
end do;

#SET: Inclusion-Exclusion

a:={1,2,3,4}; b:={1,4,6,8}; c:={1,2,3,7};

nops(a union b union c);
nops(a)+nops(b)+nops(c)-nops(a intersect b)-nops(a intersect c)-nops(b intersect c)+nops(a intersect b intersect c);

#LIST: Converting sequence to list:

M:=[s];

#TABLE: Converting list to table:

table(M);

#ARRAY:  

C:=array(1..2,1..3):
for i from 1 to 3 do C[1,i]:=i: od:
for i from 1 to 3 do C[2,i]:=i+3 od:

op(C);

whattype(C);

#-----------------------------

#Summation and Product:

i:='i';
Sum(f(i), i=1..n);
sum(i^3, i=1..10);
Product(q^i+2, i=1..4);
value(%);


#Limits, Differentiation,Extrema:
Limit((f(x+h)-f(x))/(h), x=h);
Limit(x/(x-3), x=3, right);

f:=1/(sqrt(x^3+3));
diff(f, x);
g:=cos(x);
h=(f@g)(x);
diff(h, x);

x:='x';
maximize(sin(2*x)+cos(x));
maximize(sin(2*x)+cos(x), x=0..2);
#THIS APPEARS TO HAVE CHANGED FROM WHEN THE MANUAL WAS WRITTEN

readlib(extrema):
f:=2*x^2+y+y^2;
g:=x^3-2;
extrema(f, g, {x,y});

#Integration & Related Techniques:

Int(x^3/sqrt(1-x^4), x);
value(%);

Int(e^(-x^2), x=0..infinity);
value(%);

with(IntegrationTools):
Int(x*cos(5*x), x);
Parts(%, x);
#MAY ALSO BE DIFFERENT FROM MANUAL

#Partial Fractions

rat:=(6*x^8+4*x^3-2*x^2+x-2)/(x+1)/(x^2-x+1)^2:
convert(rat, parfrac, x);

#Taylor Series

y:=1/sqrt(1-x^2);
taylor(y,x=0,9);
time(%);


#------------------------------

#2: Read Lecture Note on Euler's Theorem and RSA: a large-number example

with(NumberTheory):

e:='e';
p:=ithprime(300);
q:=ithprime(500);
e:=ithprime(200);
n:=p*q;
P:=Totient(n);
op(P);
gcd(P, e);


#these are relatively prime
#want to find d such that de == 1 mod P

d:=1/e mod P;

#suppose encrypted message is 2974943580924492084.

M:=2974943580924492084;
gcd(M, n);
M&^d mod n;

#The decrypted message is 5651777

#------------------------------

#3: Factorization Algorithm and Time Comparison

Help:=proc(): print(`MyIfactor(n)`): end:

#MyIfactor(n): inputs an integer n and returns its prime factors
#with primes in weakly increasing order

with(NumberTheory):
beep;

MyIfactor:=proc(n) local i:
i:=2;
while i<=n do
 if n mod i =0 and isprime(i) then
   return i;
 else
   i:=nextprime(i);
 fi:
 od:
end:


PrimeFacs:=proc(n) local fac, p, k;
fac:=[];
#dummy variable k is important for the iteration step below
k:=n;
while k>1 do
 p:=MyIfactor(k);
 fac:=[op(fac), p];
 k:=k/p;
od:
fac;

end: