Exercise 3.1

Which number is the largest 1000¹⁰⁰¹ or 1001¹⁰⁰⁰?

sage: 1000^1001 > 1001^1000
True

Exercise 3.2

Let us consider the following code:

sage: a = # enter a value for a
....: if a != 2:
....:     print('lost')
....: elif a == 3:
....:     print('an instant, please')
....: else:
....:     print('you win')

What is the above program doing

• when the variable a is 1?

sage: a = 1# enter a value for a
....: if a != 2:
....:     print('lost')
....: elif a == 3:
....:     print('an instant, please')
....: else:
....:     print('you win')
lost

• when the variable a is 2?

sage: a = 2# enter a value for a
....: if a != 2:
....:     print('lost')
....: elif a == 3:
....:     print('an instant, please')
....: else:
....:     print('you win')
you win

• when the variable a is 3?

sage: a = 3# enter a value for a
....: if a != 2:
....:     print('lost')
....: elif a == 3:
....:     print('an instant, please')
....: else:
....:     print('you win')
lost

• when the variable a is 15?

sage: a = 15# enter a value for a
....: if a != 2:
....:     print('lost')
....: elif a == 3:
....:     print('an instant, please')
....: else:
....:     print('you win')
lost

Exercise 3.3

Two prime numbers p and q are said twin if q = p + 2. Find all twin prime numbers below 10000.

sage: TwinPrimeNumbers = [];
....: for p in prime_range(2,10000):
....:     if (p+2).is_prime():
....:         TwinPrimeNumbers.append((p,p+2))
....: TwinPrimeNumbers

sage: len(TwinPrimeNumbers)
205

Exercise 3.4

Find the smallest and largest integers in the set

sage: min([a^b-b^a for a in range(6) for b in range(6)])
-399

sage: max([a^b-b^a for a in range(6) for b in range(6)])
399

Exercise 3.5

Recall that the method digits of an integer returns the list of its digits:

sage: 1527.digits()
[7, 2, 5, 1]

Solve Euler problem 56 by finding the maximal sum of digits of numbers of the form a^b with both a and b lesser than 100

sage: MaximalSumDigits = 0;
....: for a in srange(100):
....:     for b in srange(100):
....:         Sum = sum((a^b).digits())
....:         if MaximalSumDigits < Sum:
....:             MaximalSumDigits = Sum
....: MaximalSumDigits
972

Exercise 3.6

Solve Euler problem 4 about palindromes. A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.

Find the largest palindrome made from the product of two 3-digit numbers.

sage: LargestPalindrome = 0;
....: for a in srange(100,1000):
....:     for b in srange(a,1000):
....:         if Word((a*b).digits()).is_palindrome():
....:             LargestPalindrome = max(LargestPalindrome,a*b)
....: LargestPalindrome
906609

Exercise 3.7

Let us consider the following list of integers:

sage: l = [123, 414, 264, 18, 689, 21, 5571, 28, 589, 12, 111, 231,
....: 158, 551, 250, 68, 5728, 2222, 4198, 571, 28, 518, 999, 444,
....: 112, 689, 672, 334, 680, 273]

Construct two lists leven and lodd that contain respectively the even and odd elements of l.

sage: leven = []; lodd=[]
....: for x in l:
....:     if x%2 == 0:
....:         leven.append(x)
....:     else:
....:         lodd.append(x)

Exercise 3.8

Using an if statement involving in inside a for loop, count the number of vowels in the string:

sage: s = 'How many vowels are present in this sentence?'

sage: VowelsCount = 0
....: for i in range(0,len(s)):
....:     if s[i] in 'aeiouy':
....:         VowelsCount = VowelsCount + 1
....: VowelsCount
14

Count the number of consonant in the string:

sage: s = 'How many consonants are present in this sentence?'

sage: ConsonantsCount = 0
....: for i in range(0,len(s)):
....:     if s[i] not in 'aeiou ?':
....:         ConsonantsCount = ConsonantsCount + 1
....: ConsonantsCount
27

Exercise 3.9 (Pythagorean triples)

A Pythagorean triple is a triple (a, b, c) of positive integers so that a² + b² = c². An example is 3² + 4² = 5². How many Pythagorean triples are there with a, b and c smaller than 100?

sage: PythagoreanTripleCount = 0
....: for a in range(100):
....:     for b in range(a,100):
....:         for c in range(b,100):
....:             if a^2+b^2 == c^2:
....:                 PythagoreanTripleCount += 1
....: PythagoreanTripleCount
150

Solve Euler problem by finding the unique Pythagorean triple so that a + b + c = 1000

sage: for a in range(1000):
....:     for b in range(a,1000):
....:         for c in range(b,1000):
....:             if a^2+b^2 == c^2 and a+b+c==1000:
....:                 PythagoreanTriple = (a,b,c)
....: PythagoreanTriple
(200, 375, 425)

Exercise 3.10

Let us call a positive integer n a triple twin if all of n, n+2 and n+6 are primes. How many triple twins are there smaller than 10000?

sage: TripleTwins = 0;
....: for p in prime_range(2,10000):
....:     if (p+2).is_prime() and (p+6).is_prime():
....:         TripleTwins += 1
....: TripleTwins
55

The operator not is used for negation of a condition.

sage: not True
False

sage: not False
True