446 Examples Spring 2019.sagews

help()

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is_prime?

File: /ext/sage/sage-8.4_1804/local/lib/python2.7/site-packages/sage/arith/misc.py
Signature : is_prime(n)
Docstring :
Return "True" if n is a prime number, and "False" otherwise.

Use a provable primality test or a strong pseudo-primality test
depending on the global "arithmetic proof flag".

INPUT:

* "n" - the object for which to determine primality

See also:

  * "is_pseudoprime()"

  * "sage.rings.integer.Integer.is_prime()"

AUTHORS:

* Kevin Stueve [e-mail] (2010-01-17): delegated calculation
  to "n.is_prime()"

EXAMPLES:

   sage: is_prime(389)
   True
   sage: is_prime(2000)
   False
   sage: is_prime(2)
   True
   sage: is_prime(-1)
   False
   sage: is_prime(1)
   False
   sage: is_prime(-2)
   False

   sage: a = 2**2048 + 981
   sage: is_prime(a)    # not tested - takes ~ 1min
   sage: proof.arithmetic(False)
   sage: is_prime(a)    # instantaneous!
   True
   sage: proof.arithmetic(True)

# test if a number is prime
is_prime(5)

True

#arithemtic
print 2^3 + 2 * 5 - 3
print "2^(2^3) equals",2^(2^3)

15 2^(2^3) equals 256

#list of primes
list = prime_range(100)
print "There are",len(list),"primes less than 100"

There are 25 primes less than 100

#a way to access things in a list
for n in list:
    if n% 4 == 1:
        print n

5 13 17 29 37 41 53 61 73 89 97

# building a new list
list2 = [ p^2 for p in list]
list2

[4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409]

#how many times does 7921 appear in list2?

list2.count(7921)

1

#do arithmetic modulo n
#find an inverse of 5 moduloe 7

Mod(5,7)^(-1)

3

R = Integers(7)

for a in R:
    print a,a^2

0 0 1 1 2 4 3 2 4 2 5 4 6 1

#Solving Linear Diophantine Equations
#xgcd returns gcd(a,b), plus x and y that solve ax+by = gcd(a,b)

xgcd(5,7)

(1, 3, -2)

5*3 + 7 * (-2)

1

#x \equiv a \pmod{n} and x \equiv b \pmod{m}
#crt(a,b,n,m)
crt(5,6,11,13)

71

#squares modulo n

is_square(Mod(2^2,101))

True

is_square(Mod(2,101))

False

#primitive roots and order

Mod(2,101).multiplicative_order()

100

primitive_root(101)

2

def encode(s):
    s = str(s)
    return sum(ord(s[i])*256^i for i in range(len(s)))

def decode(n):
    n = Integer(n)
    v= []
    while n!=0 :
        v.append(chr(n %256))
        n//= 256 # replaces n by floor(n/256)
    return ''.join(v)

m = encode('I like number theory')
m

693339872193417233970224178043240443845515419721

decode(m)

'I like number theory'

#large powers, efficiently
Mod(3,10003)^10454838383

341

#Gaussian Integers
ZZ[I]

Order in Number Field in I with defining polynomial x^2 + 1

a = ZZ[I](5)
factor(a)

(I) * (-I - 2) * (2*I + 1)

(2 + I) *(3-I)

I + 7

continued_fraction(e)

[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]

cf = continued_fraction([2,5,4,3,2,7])
cf.convergents()

[2, 11/5, 46/21, 149/68, 344/157, 2557/1167]