This set is due February 6 at the beginning of lecture. Consult the syllabus for details on the homework policy.
1. a. Complete the proof by induction that if are integers and , then for all integers .
b. This result allows us to give a nice proof of the following fact: Let be a natural number and let be a positive integer. If the -th root of , , is rational, then it is in fact an integer. (The book gives a proof of a weaker fact.) Prove this result as follows: First verify that if and , then . Show that any fraction with integers, can be reduced so . Assume that is rational, say . Then also . Express this last fraction as a rational number in terms of . Use that for all and the general remarks mentioned above, to show that is in fact an integer.
2. Show by induction that for all integers there is a polynomial with rational coefficients, of degree and leading coefficient , such that for all integers , we have . There are many ways to prove this result. Here is one possible suggestion: Consider .
3. Euclidean algorithm. We can compute the gcd of two integers , not both zero, as follows; this method comes from Euclid and is probably the earliest recorded algorithm. Fist, we may assume that are positive, since , and also we may assume that , so . Now define a sequence of natural numbers as follows:
- , .
- Given , if , then .
- Otherwise, , and we can use the division algorithm to find unique integers with such that . Set .
- Let be the set of those that are strictly positive. This set has a least element, say . By the way the algorithm is designed, this means that .
- Show that , and that we can find from the algorithm, integers such that .
(If the description above confuses you, it may be useful to see the example in the book.)
4. Assume that the application of the algorithm, starting with positive integers , takes steps. [For example, if and , the algorithm gives:
Step 1: , so .
Step 2: , so .
Step 3: , so , and . Here, ]
Show that , where the numbers are the Fibonacci numbers, see Exercises 15-22 in Chapter 1 of the book.
5. Extra credit problem. With as in the previous exercise, let be the number of digits of (written in base 10). Show that