2446.sagews

"""
SAGE has a large list of built-in functions and a facility for user defined functions.
In this worksheet we shall see how to access the built-in functions, how to add user defined
functions and how to see more information about them.

A list of user defined functions can be found in the SAGE REFERENCE MANUAL.
"""

#Built-in function: a demonstration.
is_prime?

File: /usr/local/sage2/local/lib/python2.6/site-packages/sage/rings/arith.py

Type: <type ‘function’>

Definition: is_prime(n)

Docstring:

Returns True if n is prime, and False otherwise.

AUTHORS:

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

INPUT:

• n - the object for which to determine primality

OUTPUT:

• bool - True or False

EXAMPLES:

sage: is_prime(389)
True
sage: is_prime(2000)
False
sage: is_prime(2)
True
sage: is_prime(-1)
False
sage: factor(-6)
-1 * 2 * 3
sage: is_prime(1)
False
sage: is_prime(-2)
False

ALGORITHM:

Calculation is delegated to the n.is_prime() method, or in special cases (e.g., Python int``s) to ``Integer(n).is_prime(). If an n.is_prime() method is not available, it otherwise raises a TypeError.

"""
Help returns four fields:
    File:   name of file implementing the object
    Type:   type of object
    Definition: Shows how the function is called
    Docstring: Displays the documentation string that was placed in the source code.
"""

a = factorial(90) + 1
is_prime(a)

False

#How big is 90!+1?
len(str(a))

139

#A sample of functions, syntax, recursion and more.
def example1(num1, num2):
    """
    Returns the product of the numbers
    """
    ans = num1*num2
    return ans

a = example1(87,5)

example1?

File: /tmp/tmpYyP2xw/___code___.py

Type: <type 'function'>

Definition: example1(num1, num2)

Docstring:

    Returns the product of the numbers

print a

435

def example2(num1):
    """
    Prints the Fibbonaci numbers smaller than num1
    """
    a, b = 0, 1 #multiple assignments a = 0, b = 1
    while b < num1:
        print b, 
        #with the "," it prints on one line; without it is printed on  separate lines    
        a, b = b, a+b

example2(30)

1 1 2 3 5 8 13 21

example2(980)

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610

def example3(num1, num2):
    """
    Returns the largest number. Indentation critically important!
    """
    if num1 > num2:
        ans = num1
    else:
        ans = num2
    return ans

aq = example3(106, 89)
print aq

106

#Just another call
b1 = example3(23, example3(45,12))
print b1

45

#A recursive example
def mygcd(a,b):
    """
    Returns the greatest common divisor of the integers a and b
    """
    if a*b == 0: 
        answer = a + b
    else:
        answer = mygcd(b, a%b)
    return answer

fd = mygcd(879, 360)
print fd

3

#In how many ways can you write n as a sum of k positive integers
def sumofIntegers(n,k):
    """
    Returns the number of different ways the positive integer n\n 
    can be expressed as a sum of k positive integers.
    """
    if (k > n) or (k <= 0):
        ans = 0
    elif (k == 1) or (k == n):
        ans = 1
    else:
        ans = sumofIntegers(n-k, k) + sumofIntegers(n-1, k-1)
    return ans

x = sumofIntegers(11,3)

print x

10

x = sumofIntegers(11,4)
print x

11

y = sumofIntegers(23, 4)
print y

94

a = 2^127 + 1
is_prime(a)

False

def IntSize(n):
    """
    Returns the number of digits in the decimal representation of the positive integer n.
    """
    k = 0
    while n > 0:
        k = k + 1
        n = (n - n%10)/10
    return k

IntSize(factorial(90))

139

x = factorial(90) + 1
is_prime(x)

False

def mystery(n):
    """
    returns prime greater or equal to n
    """
    if is_prime(n):
         return n
    return mystery(n+1)

mystery(9)

11

mystery(6000)

6007

#booleans
a = true; b = false; a and b; a or b;

False True

for x in (false, true):
    for y in (false, true):
        if (x and not y):
            print "x = ",x,"y=",y

x = True y= False

#Boolean functions
def  nand(a,b):
    x = not (a and b)
    return x

a = true; b = false; nand(a,b)

True

nand(a,a)

False

nand(b,b)

True

#the xor function
def  xor(a,b):
    x = (a == b)
    return not x

xor(a,b)

True

#The implies function
def implies(a,b):
    x = true
    if (a == true) and (b == false):
        x = false
    return x

implies(a,b); implies(b,a)

False True