sage drill

q1.sage

K = 214805
P = Primes()
p, q = P.next(K^8+1), P.next(K^8+K^4+K+1)

x = crt(1, 2, p, q)
print x

q2.sage

K = 214805
P = Primes()

p = 10^200 + K * 10^10 + 1
while(p not in P):
  p += 10^20

print p

q3.sage

K = 214805
m = K^20
P = Primes()

p = m + 1
while(not(p in P and p+2 in P)):
  p = P.next(p)

print p

q4.sage

# http://www.sagemath.org/doc/constructions/polynomials.html

K = 214805
gf = GF(2**128, 'x')
R = PolynomialRing(gf, 'x')
x = R.gen()
S = R.quotient(gf.modulus(), 'a')
a = S.gen()

sum, pow = 0, 1
for i in (a^K).list():
  if i == 1:
    sum += pow
  pow *= 2

print sum

q5.sage

K = 214805
p = 2^192 - 2^64 - 1
seed = K^8 + K^5 + K + 1

b = p - 1
a = Mod(seed * b^2, p).nth_root(3)

x = 1
while(Mod((x^3 + a * x + b)^((p-1)/2), p) != 1):
  x += 1

y = Mod(x^3 + a*x + b, p).nth_root(2)
if(Mod(y, 2) == 1):
  y = Mod(-y, p)

xP, yP, xQ, yQ = x, y, 0, Mod(b**0.5, p)
for _ in range(10):
  L = (yQ - yP)/(xQ - xP)
  xR = L^2 - xP - xQ
  yR = L * (xP - xR) - yP
  xQ = xR
  yQ = yR

# print Mod(xR^3 + a*xR + b, p) == yR^2
print "(%d, %d)" % (xR, yR)