The construction of the Pathway

Have you attended a mathematics lecture, followed each step of the argument, and yet at the end felt that you did not understand what it was about? Have you read a proof of a theorem in a book and felt the same? If so, you have experienced a feeling common to most mathematicians.

This book on number theory has been put together by keeping a record of how I actually resolved the blocks which I encountered as I read a number of standard texts. Time and again, it was the exploration of special cases which illuminated the generalities for me. This collection of explorations was then organised into a sequence in such a way that the ‘pathway’ would climb towards the standard theorems which occur here as problems for the student at the end of each section.

The motivation for assembling the Pathway was a college need to mount a course for which lectures would not be given. If the Pathway is more successful than some other books or undergraduate lecture courses in number theory, it is because it follows more closely than usual the natural process of discovery, and puts logic in its proper place. The purpose of rigour', said Hadamard, ‘is to legitimate the conquests of the intuition, and it has never had any other purpose’.

The construction of the Pathway

Have you attended a mathematics lecture, followed each step of the argument, and yet at the end felt that you did not understand what it was about? Have you read a proof of a theorem in a book and felt the same? If so, you have experienced a feeling common to most mathematicians.

This book on number theory has been put together by keeping a record of how I actually resolved the blocks which I encountered as I read a number of standard texts. Time and again, it was the exploration of special cases which illuminated the generalities for me. This collection of explorations was then organised into a sequence in such a way that the ‘pathway’ would climb towards the standard theorems which occur here as problems for the student at the end of each section.

The motivation for assembling the Pathway was a college need to mount a course for which lectures would not be given. If the Pathway is more successful than some other books or undergraduate lecture courses in number theory, it is because it follows more closely than usual the natural process of discovery, and puts logic in its proper place. The purpose of rigour', said Hadamard, ‘is to legitimate the conquests of the intuition, and it has never had any other purpose'. Formality, abstraction and generality have an essential place in the completion of any piece of mathematics, but their role in discovery is varied. In the Pathway, the introduction to each new idea is as informal and as specific as I could make it. There are a few remarks about foundations in the notes, but the statement of the Peano axioms postdates almost all of the number theory in the book and is not given here.

In the selection of material, the needs of future teachers have been kept in mind. The theme of sums of squares links chapters 4, 5, 6, 8, 9 and 10. Arithmetic functions and the distribution of primes were thought to offer less connection with school work.

Unimodular transformations

We start by examining the transformations of a square lattice onto itself.

1 On some square lattice paper, choose four lattice points A, B, C and D so that ABCD forms a parallelogram, ᴨ, without lattice points inside the parallelogram or on its perimeter except at its vertices. Sketch the image of ᴨ under the translation of the lattice which maps A to B. Call this translation. Sketch the image of ᴨ under the translation of the lattice which maps A to D. Call this translation σ. Do and σ (ᴨ) have lattice points inside them or on their perimeters? Do or have lattice points inside them or on their perimeters for any integers m or nl Is the same true for the parallelogram?

2 With the notation of q 1, does every point of the plane lie within or on the perimeter of one or more of the parallelograms? Must each lattice point be a vertex of four such parallelograms?

3 Using the conventional coordinate system with rectangular cartesian axes for the plane, the set of square lattice points may be labelled with the set of all ordered pairs of integers. If ABCD is a parallelogram of lattice points as in q 1, we are free to choose A as the origin of our coordinate system. If B = (a, b) and D = (c, d), what are the coordinates of C if C is the vertex of the parallelogram opposite Al What is the image of the unit square ﹛(0, 0), (1, 0), (1, 1), (0,1)﹜ under the linear transformation

What are the images of the lattice points of the form (x, 0) under α? What are the images of the lattice points of the form (0, y) under α ? The lines x = k, y = l, parallel to the axes, with k and l integers, form a grid of unit squares. What is the image of this grid under the linear transformation α?

4 Does the linear transformation α of q 3, map the set of lattice points onto itself?

Division algorithm

• 1 Look at table 1.1. If the same pattern was extended downwards, would it eventually incorporate any positive integer ﹛1, 2, 3, …, n, n +1,…﹜ that we might care to name?

• 2 What is the relation between each number in table 1.1 and the number below it?

• 3 Give a succinct description of the full set of numbers in the column below 0.

• 4 If you choose two numbers from the column below 0 and add them together, where in the table must their sum lie?

• 5 The whole of the array in table 1.1 may be considered as an addition table with the column below 0 down one side and the numbers 1, 2, 3 across the top. Using your brief description of the numbers in the column below 0, devise a comparably succinct description of the full set of numbers in the column below 1, and similarly succinct descriptions of the sets of numbers in the other two columns.

• 6 If two numbers lie in the second column and the lesser is subtracted from the greater, where does the difference lie?

• 7 If two numbers lie in the third column and the lesser is subtracted from the greater, where does the difference lie? Try to prove your result in a general way which would apply to all such pairs.

• 8 If two numbers lie in the fourth column and the lesser is subtracted from the greater, where does the difference lie? Prove it.

• 9 If two numbers are chosen, both from the second column, where does their sum lie? Prove your claim generally.

• 10 If two numbers are chosen, both from the fourth column in table 1.1, where does their sum lie? Prove your claim generally.

• 11 Are there general rules which enable you to fill in the table below for addition of numbers by columns? If only the numbers at the heads of the columns are used in this table, the table that results is an example of an addition table modulo 4. Such a table is denoted by (Z4, +).

Ferrers’ graphs

1 1+1+1+1=2+1+1=2+2=3+1+4.

Because of this, the positive integer 4 is said to have five partitions. Exhibit the seven partitions of 5 and the eleven partitions of 6. How many partitions of 7 are there?

2 The five partitions of 4 may be exhibited thus:

and these are called the graphs of the partitions. Exhibit the graphs of the partitions of 5 and of 6.

3 The columns of the graph : • • give the partition 2 + 1 + 1. The rows of the same graph give the partition 3 + 1. Interchanging rows and columns converts : • • to : •. Such pairs of partitions are said to be conjugate. Which partition of 4 is not conjugate to any other partition of 4? Suggest a name for such a partition. Find all the partitions of numbers up to 10 which are not conjugate to any other partition.

4 Exhibit all the partitions of the numbers up to 10 into distinct odd parts.

5 Does every odd number have at least one self-conjugate partition?

6 If a number has a self-conjugate partition, must it have at least one partition into distinct odd parts?

7 If a number has a partition into distinct odd parts, must it have a self-conjugate partition?

8 Conjecture and prove a theorem about the numbers of selfconjugate partitions of a number n and the number of partitions of n into distinct odd parts.

9 Exhibit the partitions of the numbers from 1 to 7 which have only the numbers 1 and/or 2 as parts.

10 How many partitions of 2n are there which have only the numbers 1 and/or 2 as parts?

How many partitions of 2n+1 are there which have only the numbers 1 and/or 2 as parts?

The number of partitions of n in which each part is 1 or 2 is denoted by P2(n).

11 Exhibit the partitions of the numbers from 1 to 7 which have only one or two not necessarily distinct parts.

12 How many partitions of 2n are there which have only one or two not necessarily distinct parts?

What is the number of partitions of 2n+1 which have only one or two not necessarily distinct parts?

A pocket calculator is needed for this and the following chapter.

Irrational square roots

1 If a and b are positive real numbers and a/b = √2, prove that (2b -a)/(a-b) = √2 and that b > a-b > 0. Use Fermat's method of descent to prove that a and b cannot both be integers.

2 If a and b are positive real numbers and a/b =√7, prove that (7b-2a)/(a-2b) = √7 and that b>a-2b>0. Deduce that a and b cannot both be integers.

3 Construct a proof, similar to the ones you have used in q 1 and q 2, which establishes that √57 is not a rational number.

4 Find integers a, b, c, d such that 2520/735 = 2a3b5c7d.

5 If p1 p2, P3,… is the sequence of distinct prime numbers, explain why every non-zero rational number can be expressed in the form for some n and uniquely defined integers a, a2, …, an.

6 If what can be said about the indices for r2?

7 Can 2, 3, 5 or 6 be the square of a rational number?

Convergence

The notion of a continued fraction emerges from an attempt to find rational approximations to irrational square roots.

8 Prove that, and deduce that

9 The five numbers

are also conventionally exhibited in the form

and in the form

[1], [1,2], [1,2,2], [1,2,2,2], [1,2,2,2,2].

• (i) Express each of these five numbers as a quotient of integers.

• (ii) If a/b and c/d are adjacent terms in (i), find ad-be for the four possible cases.

• (iii) Use a calculator to express these five numbers in decimal form, and notice whether the sequence increases, decreases or oscillates, and whether it appears to approach √2.

10 For real numbers a1, a2, a3,…, an, the simple continued fraction is denoted by [a1, a2, a3, …, an]. Convention demands that a, when ai≥2 .

Sums of two squares

1 Determine the positions in table 1.1 of the integers less than 100 which appear in table 5.1.

2 Which columns of table 1.1 contain numbers from table 5.1 and which do not?

3 Which columns of table 1.1 contain odd prime numbers from table 5.1 and which do not?

4 Make a table of sums of squares modulo 4 and prove that no integer of the form 4k + 3 can be a sum of two squares.

5 Make a list of those numbers less than 200 in tables 5.1 which have a factor 3. Determine, in each case, the highest power of 3 which divides the number.

6 Make a table of sums of squares modulo 3. What can you deduce about x and y if x2 + y2 ≡ 0 (mod 3)? And what does this imply about the number x2 + y2?

7 Make a list of those numbers less than 200 in table 5.1 which have a factor of 7. Determine, in each case, the highest power of 7 which divides the number.

8 Make a table of sums of squares modulo 7. What can you deduce about x and y if x2 + y2 = 0 (mod 7)? What does this imply about the number x2 + y2?

9 Suppose a prime number p, different from 2, divides a number of the form x2 + y2, so that x2 + y2 = 0 (modp); then if y ≢0 (modp), there is an integer a such that ay = \ (mod p), so that ﹛ax)2 +1 = 0 (mod p). Thus ax is an element of order 4 in Mp. What does Lagrange's theorem on subgroups now tell you about p? (Compare with q4.18 and q4.19.)

10 Try formulating a contrapositive to q 9 with a view to generalising q 6 and q 8.

11 Explore the validity of the argument in q 9 when p = 2.

12 If x2 + y2 is divisible by 27, must it be divisible by 81? Generalise your argument.

13 Examine the prime factorisation of numbers in table 5.1 with a view to making a conjecture about which positive integers may be expressed as a sum of two squares.

Congruence classes and the Chinese remainder theorem

1 If a and b are numbers in table 2.1 and you are given that 6 is a factor of a—by what can you say about the positions of a and b in the table?

2 If 6 is a factor of a — b, we write

a = b (mod 6)

and say that a is congruent to b modulo 6.

Explain why a = a (mod 6) for all integers a.

If a ss b (mod 6), prove that b ss a (mod 6).

If a ss b (mod 6) and b = c (mod 6), prove that a = c (mod 6).

3 Determine the set of all integers congruent to 0 (mod 6).

Determine the set of all integers congruent to 1 (mod 6).

Determine the set of all integers congruent to 2 (mod 6).

Determine the set of all integers congruent to 3 (mod 6).

Determine the set of all integers congruent to 4 (mod 6).

Determine the set of all integers congruent to 5 (mod 6).

These six sets are called the congruence classes or residue classes modulo 6. Use the division algorithm to prove that each integer belongs to exactly one of these classes.

4 When n is a factor of a-b, we write

a = b (mod n)

and say that a is congruent to b modulo n.

We presume here that a, b and n are integers, and n is positive.

State and prove generalised forms of q 2 and q 3 for integers modulo n.

5 By using tracing paper or by making a copy of the table, enter the numbers 0, 1, 2, 3, 4 and 5 in the appropriate square of the array given below. For example, 4 = 0 (mod 2) and 4 = 1 (mod 3).

6 By using tracing paper or by making a copy of the table, enter the numbers 0, 1, 2, …, 9, 10, 11 in the appropriate square of the array given below.

7 By using tracing paper or by making a copy of the table, enter the numbers 0, 1, 2, …, 33, 34, 35 in the appropriate square of the array given below.

Fermat's theorem

1 6 ≡ 14 (mod 4) and 3 ≡ -1 (mod 4). Is it true that 6 • 3 ≡14(-1) (mod 4)?

If a ≡ b (mod 4) and c ≡ d (mod 4), can you prove that ac ≡ bd (mod 4)?

2 Make a multiplication table modulo 4 for the complete set of residues ﹛-4, 5,2, -1﹜. Does it have the same structure as the table you found for q 1.14?

3 If a ≡ b (mod n) and c ≡ d (mod n), use the equality ac-bd≡ (a-b)c + b(c-d) to prove that ac ≡ bd (mod n).

4 If a1, …, an is a complete set of residues modulo n, and the product modulo n of ai and aj is defined to be ak where aiaj (mod n), is ak uniquely defined?

Is there an element az such that a-za≡az for all i?

Is there an element aI such that aIai ≡ ai for all i?

5 If a1 …, an is a complete set of residues modulo n, and bu • • •, bn is a complete set of residues modulo n, how would you establish that the tables of products modulo n obtained from these two sets have the same structure? Your answer establishes a well-defined table for (Zn, x).

6 Examine the tables for multiplication modulo 3, 4, …, 16 given in table 3.1. Is there an identity in (Zn, x), for every n?

7 Compare the second row of the tables for (Z3, x), (Z4, x), …, (Z1o, x) with your answers to q 2.28. What is the similarity?

8 Compare the third row of the tables for (Z3, x), …, (Z10, x) with your answers to q 2.29. What is the similarity?

9 Compare the fourth row of the tables for (Z3, x), …, (Z10, x) with your answers to q 2.30. What is the similarity?

Quadratic residues and the Legendre symbol

1 Working in Z7, find all the solutions to x2 = 0, 1, 2, 3, 4, 5, 6 respectively.

2 In Z7, which of the equations

have no solution, have one solution, have two solutions?

3 The perfect squares in M7 are called the quadratic residues modulo 7. Exhibit the mapping of M7 given by x → x2. Does the set of images under this mapping form a subgroup of M7?

4 Use table 3.2 to find the quadratic residues modulo 5, 11, 12 and 13. For which of these moduli do the quadratic residues form a subgroup of Mn ? For which of these values of n is the mapping of Mn given by x → x2, two to one?

5 State the number of quadratic residues modulo 3, 5, 7, 11, 13 and 17 respectively. Predict the number of quadratic residues modulo p (an odd prime).

6 If x2 ≡ y2 (mod p), does it follow that either x = y (mod p) or x =-y (modp), when p is a prime number?

7 For any prime p, determine the elements of Mp which are mapped to 1 under the mapping x → x2. Can you say how many elements of Mp are mapped to any other quadratic residue under this mapping?

8 A number in Mn which is not a square is called a quadratic nonresidue modulo n. Find three quadratic non-residues modulo 12.

9 Find the cube of all the quadratic residues modulo 7, and also the cube of all the quadratic non-residues.

10 What does Fermat's theorem imply about the number of roots of x6=1 (mod 7)?

What can you deduce about the cube of a quadratic residue modulo 7?

By factorising x6- 1 and using Lagrange's theorem, q 3.57, deduce that at most three cubes are congruent to 1 (mod 7) and at most three cubes are congruent to —1 (mod 7).

11 Using the method of q 10, determine how many fifth powers are congruent to 1 (mod 11) and how many fifth powers are congruent to-1 (mod 11). Does the value of the fifth power determine whether the number is a quadratic residue or not?