Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
Summary
Sums of two squares
Determine the positions in table 1.1 of the integers less than 100 which appear in table 5.1.
Which columns of table 1.1 contain numbers from table 5.1 and which do not?
Which columns of table 1.1 contain odd prime numbers from table 5.1 and which do not?
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.
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.
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 *2 + y2?
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.
Make a table of sums of squares modulo 7. What can you deduce about x and y if *2 + y2 = 0 (mod 7)? What does this imply about the number *2 + y2?
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- A Pathway Into Number Theory , pp. 119 - 139Publisher: Cambridge University PressPrint publication year: 1996
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