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Section3.6Exercises

1

For each of the following linear Diophantine equations, either find the form of a general solution, or show there are no integral solutions.

  • \(21x+14y=147\)

  • \(30x+47y=-11\)

3

Find all simultaneous integer solutions to the following system of equations. (Hint: do what you would ordinarily do in high school algebra or linear algebra! Then finish the solution as we have done.)

  • \(x+y+z=100\)

  • \(x+8y+50z=156\)

4

Compute the number of positive solutions to the linear Diophantine equation \(6x+9y=c\) for various values of \(c\) and compare to the analysis we did above.

5

Explore the patterns in the positive integer solutions to \(ax+by=c\) situation above. For sure I want you to do this for the ones I mention there, but try some others and see if you see any broader patterns!

6

Prove that any line \(ax+by=c\) which hits the integer lattice but \(\gcd(a,b)\neq 1\) is the same as a line \(a'x+b'y=c'\) for which \(\gcd(a',b')=1\), and explain why that means that without loss of generality the first topic doesn't need any more explanations.

7

Find a primitive Pythagorean triple with at least three digits for each side.

8

Prove that 360 cannot be the area of a primitive Pythagorean triple triangle.

9

Find a way to prove that \(x^4+y^4=z^4\) is not possible for any three positive integers \(x,y,z\). (Hint: use Corollary 3.4.12; this one is harder.)

10

We already saw that if \(x,y,z\) is a primitive Pythagorean triple, then exactly one of \(x,y\) is even (divisible by 2). Assume that it's \(y\), and then prove that \(y\) is divisible by 4.

11

Under the same assumptions as in the previous problem, prove that exactly one of \(x,y,z\) is divisible by 3. (Combined with the previous exercise, this proves that every area of a Pythagorean triple triangle is divisible by 6. Is it also true that exactly one of \(x,y,z\) is divisible by 5?)

12

A Pythagorean triple satisfies \(x^2+y^2=z^2\). Explore patterns for triples of positive integers which satisfy \(x^2-xy+y^2=z^2\). If Pythagorean triples correspond to right triangles, what sort of triangles do these triples correspond to?

13

Find a (fairly) obvious solution to the equation \(m^n=n^m\) for \(m\neq n\). Are there other such solutions?

14

Show that \begin{equation*}\gcd(x,y)^2=\gcd(x^2,xy,y^2)\end{equation*} which we use in Proposition 3.7.2. You can try this using the set of divisors definition of gcd, or using the definition \(\gcd(a,b,c)=\gcd(\gcd(a,b),c)\).

15

Explore Bresenham's algorithm in print or online. What is the connection to this chapter? How do non-solutions to linear Diophantine equations relate to actual solutions, in this context?