1. Exercise 1.4.2.

2. True or False? Every integer $n>2$ occurs in some Pythagorean Triple. (Justify your answer.)

3. Explain why there are only finitely many distinct Pythagorean Triples $(a,b,c)$ with $a=100$.

Solutions.tex.
Solutions.pdf.

1. Give a geometric proof that $\sqrt{3}$ is irrational. (Hint: It might be easier to show that $1+\sqrt{3}$ is irrational, then deduce that $\sqrt{3}$ is also irrational.)

2. Use the Euclidean algorithm to find an integral solution to $270x+168y = 6$.

3. What is the height of a regular tetrahedron of side length 1?

Solutions.pdf.

1. Exercise 2.2.2.

2. Exercise 2.2.3.

3. Let $P$ be a polyhedron. Suppose $F_1, \ldots, F_k$ are the faces of $P$ that meet at vertex $V$, and that $A_1, \ldots, A_k$ are the angles of these faces at $V$. Define the defect at vertex $V$ to be ($360$-(sum of the angles $A_i$)). (For example, in a cube there are three squares meeting at any vertex, so the defect at any vertex is ($360-$($90+90+90$)) = $90$ degrees.) The total defect of $P$ is the sum of the defects at all of the vertices of $P$. Exercise: find the total defect of each of the Platonic solids.

Solution sketches.

2. Show that if a convex polygon is constructible, then its area is a constructible number. (Hint: start with triangles.)

3. Show that if a regular polygon of circumradius 1 has constructible area, then it is possible to construct a copy of the polygon. (The circumradius is the radius of the circumscribing circle.) (Hint: First show that the area of a regular $n$-gon with circumradius $1$ is $n\sin(\pi/n)\cos(\pi/n)$.)

Solution sketches.

1. Exercise 2.3.3.

2. Exercise 2.3.4. (Briefly explain the steps of the construction.)

3. Verify the correctness of the construction of the regular pentagon indicated in the gif above.

Solution sketches.

1. Find the continued fraction for $\frac{34}{21}$ and use it to find a solution to this instance of Bézout's Identity: $21x+34y=1$.

2. Use Brahmagupta's composition method to find a solution to $x^2-dy^2=1$, where $d=n^2+1$.

3. The quadratic mean of a sequence $a_1,\ldots,a_n$ is $$ \sqrt{\frac{a_1^2+\cdots+a_n^2}{n}}. $$ Find an integer $n>1$ such that the quadratic mean of the first $n$ positive integers is again an integer. That is, find $n>1$ such that $$ \sqrt{\frac{1^2+2^2+\cdots+(n-1)^2+n^2}{n}} $$ is a positive integer. (Hint: Reduce this problem to Pell's equation using the formula $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$.)

Solution sketches.

2. Given integers $a$ and $b$, are the following congruences compatible?
$x\equiv a\pmod{b}$
$x\equiv b\pmod{a}$

3. I am thinking of a number between $1$ and $1000$. I am willing to tell you the least significant digit of my number in each of the bases $2, 3, \ldots, 10$. Is this enough information to determine the number? (The least significant digit of a number written $a_na_{n-1}\cdots a_1a_0$ in base $b$ is $a_0$.)

Solution sketches.

1. Exercise 6.5.2 from the text.

2. Find all of the roots of the sextic $x^6 - 15x^2 - 4 = 0$ using an adapted form of the Cardano formula. (That is, let $y=x^2$, solve $y^3 - 15y - 4 = 0$ with the Cardano formula, then find $x$.) Which of your roots is equal to the root $x=2$?

3. Sketch the parabola $y=x^2-x$ and the hyperbola $xy=1$ together, and locate all (real) points of intersection of these curves.

Solution sketches.

1. Suppose that $f(x)$ is a 1-variable polynomial of degree greater than $1$. Explain how to find the points at infinity on the curve $y=f(x)$.

2. Is there a (complex) polynomial $F(x,y)$ such that the projective completion of the curve defined by $F(x,y)=0$ has no points at infinity?

3. Everybody knows that $2$ points are sufficient to determine a line. How many points are sufficient to determine an irreducible conic in the projective plane over $\mathbb C$?

Solution sketches.

1. Exercise 8.7.1 from the text.

2. Exercise 8.7.2 from the text.

Remark: Where the author writes: the cubic curves $y = x^3$ and $y^2 = x^3$ have the same projective completion, read this as saying the cubic curves $y = x^3$ and $y^2 = x^3$ have the same projective completion up to a linear change of coordinates.

3. Find all points of intersection of the curves $y=x^2$ and $y=x^3$, and compute the intersection multiplicity at each point of intersection.

Solution sketches.