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MATH 381: Assignment 1 (due: September 26, 2024)

1. Prove that √2 + √3 is an irrational number.

2. If (a, b, c) is a primitive Pythagorean triple so that a 2 + b 2 = c 2 , show that c cannot be even.

3. Show that the diagonals of a regular pentagon trisect the angle at each vertex.

4. Given any pentagon, show that one can construct a square using straightedge and compass, such that the area of the square is equal to the area of the given pentagon.

5. In Figure 1 below, O is the center of the circle. Show that α = 2β.

Figure 1

Figure 2

6. In Figure 2 above, O is the center of the circle. Show that α = β.

7. In Figure 3 below, AC is the diameter of the circle of radius (a + 1)/2. If DB is perpendicular to AC, show that DB has length √a.

Figure 3

8. In the figures below, calculate x and y in terms of a and b. Deduce that if a, b are constructible, then so are ab and a/b. Conclude that the set of constructible numbers forms a field.

9. If n is a composite number, show that 2 n− 1 is also a composite number.

10. Determine all natural numbers x, y, z that satisfy the equation