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

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[…] Homework 1, due February 6, at the beginning of lecture. Possibly related posts: (automatically generated)What Lil’ Ones Are Reading: Two Christmas Stories […]

Perhaps this is a stupid questions, but what does the notation (a,b) = 1 mean in the first problem? Is it a function that I have missed somewhere? Is it equivalent to a = b = 1, so that we are to prove that a = a^n = 1 and b = b = 1 for all integers n >= 1? I’m a little lost…

Hi. The notations and mean the same thing: The greatest common divisor of and .
[I agree there is a little ambiguity in using this way, but it is standard in number theory.]

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

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Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]

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This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]

[…] Homework 1, due February 6, at the beginning of lecture. Possibly related posts: (automatically generated)What Lil’ Ones Are Reading: Two Christmas Stories […]

Perhaps this is a stupid questions, but what does the notation (a,b) = 1 mean in the first problem? Is it a function that I have missed somewhere? Is it equivalent to a = b = 1, so that we are to prove that a = a^n = 1 and b = b = 1 for all integers n >= 1? I’m a little lost…

Hi. The notations and mean the same thing: The greatest common divisor of and .

[I agree there is a little ambiguity in using this way, but it is standard in number theory.]