Exercise 1.1

A partition of a nonnegative integer n is a non-increasing list of positive integers with total sum n. Does Sage have a command for defining a partition ?

sage: Partition

(hint: type Part and then hit the <TAB> key)

Create the partition with list [4, 3, 1, 1] and assign it to the Python name p :

sage: p = Partition([4, 3, 1, 1])

(hint: to see documentation and examples for the Permutation command, type Partition?)

Find the conjugate of p (and name it q):

sage: q = p.conjugate(); q
[4, 2, 2, 1]

Which of p and q dominates the other ?

sage: print p.dominates(q)
....: print q.dominates(p)
True
False

(hint: to see the methods available to the partition named p, you can type p. and hit <TAB>)

How did Sage decide whether p dominates q ?

sage: p.dominates??

(hint: to access the source code, use p??)

Exercise 1.2

Find out what is the name of the function to construct a matrix

sage: matrix([[1,2],[3,4]])
[1 2]
[3 4]

Compute the rank, the determinant and the Hermite normal form of the following 5x5 integer matrix

sage: M = matrix(5, 5, [8,9,4,6,0,0,0,9,9,8,7,6,5,4,4,5,9,3,9,5,6,4,0,3,6]); M
....: print M.rank()
....: print M.determinant()
....: print M.hermite_form()
5
-7117
[   1    0    0    0 2308]
[   0    1    0    0 6657]
[   0    0    1    0 2420]
[   0    0    0    1  744]
[   0    0    0    0 7117]

Obtain a latex code for this matrix using the function latex

sage: latex(M)
\left(\begin{array}{rrrrr}
8 & 9 & 4 & 6 & 0 \\
0 & 0 & 9 & 9 & 8 \\
7 & 6 & 5 & 4 & 4 \\
5 & 9 & 3 & 9 & 5 \\
6 & 4 & 0 & 3 & 6
\end{array}\right)

Exercise 1.3

Find out the name of the method on integers that give the list of prime factors of an integer n

sage: n.prime_factors()
[5, 2917]

(note: to do that, you first need to assign an integer to a variable and use tab-completion on this variable)

Solve Euler problem 3: what is the largest prime factor of the number 600851475143

sage: 600851475143.prime_factors()[-1]
6857

Exercise 1.4

Find the name of the function that given an integer n returns the n-th prime number

sage: primes_first_n(5)[-1]
11

Solve Euler problem 7: what is the 100001-th prime number

sage: primes_first_n(100001)[-1]
1299721

Exercise 1.5

Solve Euler problem 10: what is sum of prime numbers below two milions?

sage: sum(prime_range(2000000))
142913828922

(hint: You can use the function sum to make the sum of elements in a list.)

Exercise 1.6

What is the name of the method on integer that provides the list of digits?

sage: n = 131
....: n.digits()
[1, 3, 1]

Check that the sum of digits of 2¹⁵ = 32768 is 3 + 2 + 7 + 6 + 8 = 26

sage: n = 32768
....: sum(n.digits()) == 26
True

Solve Euler problem 16 and Euler problem 20

sage: print sum((2^1000).digits())
....: print sum((factorial(100)).digits())
1366
648

Note that the comparison operators are == (for equality), != (for difference). You already used the sign = but for another purpose: assignment of value to a variable.

Exercise 1.7

Draw a graphics of the function f(x) = sin(x) + cos(x) − xexp(x) for x between 0 and 1

sage: f(x) = sin(x) + cos(x) - x * exp(x)
....: plot(f, x, 0, 1)

Using the function find_root, obtain an approximation of the unique solution r of f(r) = 0 with r ∈ (0, 1).

sage: r = find_root(f, 0, 1); r
0.6998282883190581

Draw a graphics of f together with a red vertical line between (r,  − 1) and (r, 1)

sage: P1 = plot(f, 0, 1)
....: P2 = plot(line(((r, -1), (r, 1))))
....: P1 + P2

Exercise 1.8

What are the first 20 terms of the Taylor expansion at x = 0 of the function g(x) = 1 ⁄ √(1 − 4*x)

sage: g(x) = 1 / sqrt(1 - 4 * x)
....: g.taylor(x, 0, 19)
x |--> 35345263800*x^19 + 9075135300*x^18 + 2333606220*x^17 + 601080390*x^16 + 155117520*x^15 + 40116600*x^14 + 10400600*x^13 + 2704156*x^12 + 705432*x^11 + 184756*x^10 + 48620*x^9 + 12870*x^8 + 3432*x^7 + 924*x^6 + 252*x^5 + 70*x^4 + 20*x^3 + 6*x^2 + 2*x + 1

Check on the taylor expansion that

sage: for n in range(20):
....:     print n, binomial(2*n, n)
0 1
1 2
2 6
3 20
4 70
5 252
6 924
7 3432
8 12870
9 48620
10 184756
11 705432
12 2704156
13 10400600
14 40116600
15 155117520
16 601080390
17 2333606220
18 9075135300
19 35345263800

Exercise 1.9

How many alternating sign matrices of size 100 × 100 are there?

sage: A.cardinality()
67646598758135364929766105202064061548880635165171008431467330663074020824274350145655952261209835873471998434649026898743552719017530774467665440066378212487465276748835757320937722883516914771025941994364259074081633745194182795303792782386364156011889797702249624656293623090331365176412987169107956929893906691837790203805324848278523051892678566959524684937559468309061642907141388317464554478493799375191780485737013728816786563747745275966081016332577016766378998630659070253824266585368968212004162093058178099275098925255177150341226891129333108960702052030500172949212553682337174501590698158249096078547143949845830890554022513059455443398797554920501445575972523760007181991971164600649300008296835626471233577242420415141350372936080149629409298524932704358220893836867610327072250698806614699732606572442579714990492066999635634301119490567845001613642344010705688314276991141347998778009889027025423498310806341127678866995548401832916746383750016837512269151387212372953453219592776280544284259880868849167548141899447919253425061333222531631171613408573138332668524248821566452941896629214485113421439342359825510039552

How does it compare with the number of atoms in the universe?

Exercise 1.10

Euler problem 24: what is the millionth lexicographic permutation of {0, 1, …, 9}?

sage: p = Permutations(10)
....: p[999999]
[3, 8, 9, 4, 10, 2, 6, 5, 7, 1]