Professor Simonson History of Math Ingenuity

Fall 1997 MA 149

Miscellaneous Extra Credit (30 points)

1. Prime Numbers of the form Ax+B

It was proved (1800's) by Dirichlet that there is an infinite number of primes of the form Ax+B, if A and B have no common factor except 1, and A, B, x are integers.

Is there an infinite number of non-primes of the forms Ax+B? Does it depend on whether or not A and B have a common factor besides 1. Prove all your conjectures.

2. Fermat Primes

Fermat (1600's) once considered numbers of the form 2^x+1 where x itself is a power of 2. For example 2²+1, 2⁴+1, 2⁸+1, 2¹⁶+1etc. Fermat conjecturedthat these numbers were all prime. Euler (1700's) discovered that 2³²+1 is not prime by showing that 641 divides it evenly. Verify that the first 4 Fermat numbers are prime but that 2³²+1 is not prime.

Although no more Fermat numbers are known to be prime, Polya (1900's), a mathematician famous for his teaching and his emphasis on experimentation and guessing, proved that any pair of Fermat numbers have no common factor except 1. Explain why Polya's theorem implies that there must be an infinite number of prime numbers.

3. Spherical Geometry

In spherical geometry, we define a line to be a "great circle" of the sphere. That is, it bisects the sphere.

Define a circle on spherical geometry.

Is every circle also a line? Is every line a circle? Explain.

The circumference of a circle in Euclidean Geometry is 2¹R, where R is the radius of the circle. Investigate whether this theorem holds for spherical Geometry. Make conjectures and show examples or counterexamples.

4. Sums

a. Calculate the sum 1+1/5 + 1/25 + 1/125 + ....

b. Calculate the value of the fraction .11111111 ....

c. Calculate the value of the fraction .345345345 ....

d. Calculate the value of the sum 1 + 1/2 + 2/4 + 3/8 + 4/16 +5/32 + ....

(Hint: Use the same technique as the previous problems, but do it twice.)

e. Prove that any repeating decimal is a rational number.

5. Polyhedra

We know that it is impossible to make a regular polyhedra out of hexagons. However, we can make one out of pentagons, where three pentagons meet at each point of the polyhedra.

a. We can use Euler's Theorem (F + V - E = 2), which says that in a polyhedra the number of faces F plus the number of points (or vertices) V minus the number of edges E equals 2, to prove that in a regular polyhedra made out of pentagons, the numbers of faces F is 12.

i. First prove that the number of vertices V is 5F/3.

ii. Then prove that the number of edges E is 5F/2.

iii. Then derive the exact value for F.

b. If you allow a combination of hexagons and pentagons in a polyhedra, investigate whether you can construct any such polyhedra, and if so, describe the number of hexagons and pentagons used, and explain how they are connected to each other.