Exercises 3.6 Exercises
Exercise Group.
For each of the following linear Diophantine equations, either find the form of a general solution, or show there are no integer solutions.
7.
8.
9.
Check the details and complete the proof in Subsection 3.1.4.
10.
11.
Compute the number of positive solutions to the linear Diophantine equation for various values of and compare to the three-case analysis at the end of Subsection 3.3.2.
12.
Explore the patterns in the positive integer solutions to situation in Section 3.3. For sure I want you to do this for the ones I mention there, but try some other values of and see if you see any broader patterns!
13.
Prove that any line which hits the integer lattice but is the same as a line for which and explain why that means that without loss of generality Theorem 3.1.2 doesnβt need any more explanations.
14.
Find a primitive Pythagorean triple with at least three digits for each side.
15.
Use Proposition 3.4.9 to prove that a Pythagorean triple triangle cannot have odd area.
16.
Prove that 360 cannot be the area of a primitive Pythagorean triple triangle.
17.
Find a way to prove that is not possible for any three positive integers (Hint: use Corollary 3.4.13; this exercise needs a little cleverness.)
18.
We already saw that if is a primitive Pythagorean triple, then exactly one of is even (divisible by 2). Assume that itβs and then prove that is divisible by 4.
19.
Under the same assumptions as in the previous problem, prove that exactly one of 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 is divisible by 5?)
20.
A Pythagorean triple satisfies Explore patterns for triples of positive integers which satisfy If Pythagorean triples correspond to right triangles, what sort of triangles do these triples correspond to?
21.
22.
Show that
which we use in Proposition 3.7.2. You can try this using the set of divisors definition of gcd, or using the definition
23.
Explore Bresenhamβs 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?
24.
Assume you have relatively prime integers and a positive integer Describe all positive solutions to and use Definition 2.4.1 to find (positive) solutions to
25.
Assume are odd, coprime positive integers. Show that is a primitive Pythagorean triple, and that all such triples are generated this way. (See Remark 3.4.8.)
26.
Cultures across Eurasia have variants of the βProblem of the Hundred Fowlβ (see among others [E.5.10, Chapter 15], [E.5.1, p. 176], and [E.5.11, Section 1.1.1.3]). This one is from Abu Kamil (about 900 AD). Can you find all solutions with positive integers? What if you generalize the prices of the birds? (Finding a general solution was attempted β unsuccessfully β by Chinese mathematicians for generations.)
β14β
In [E.5.3, Section 6-4] a similar example of Abu Kamilβs with five unknowns is given, which he claimed had exactly solutions in positive integers; today such computations are of high interest in computational geometry on polytopes.
Suppose ducks cost five coins each, chickens one coin each, but one coin buys twenty sparrows. If you spend one hundred coins to purchase one hundred birds, how many of each did you buy?