Math 350 - Fall 2000 - Number Theory

Suggested presentation topics

Prime Producing Polynomials

Mathematicians have long sought mechanisms for generating primes. Euler made the observation that the polynomial f(n) = n² – n + 41 takes on prime values for n = 1, 2, ..., 40. But, as we've seen, no polynomial takes on prime values exclusively. It may come as a surprise, then, that there exists a polynomial function F(a,b,...,z), taking 26 integer-valued inputs, such that the range of F, intersected with N, is precisely the set of prime numbers (see J. P. Jones et al., Amer. Math. Monthly 83, 1976). Explain how polynomials can be constructed whose set of positive values is some desired set. What sorts of sets of positive values are possible? Is such a polynomial useful, e.g., for producing large primes?

Sunflowers and the Golden Ratio

Nature has hit upon a beautifully optimal solution to the problem of arranging seeds on a sunflower head. Placing one seed at every multiple of a fixed angle, with distance from center increasing so as to maintain a constant number of seeds per unit area, results in optimal seed dispersion when the angle is some fraction of 2*pi which involves the golden ratio 0.61803.... Explain this phenomenon. Is it surprising that the sunflowers seem to reproduce the golden ratio to such a high degree of precision? How does this tie in with Fibonacci numbers? Are there other situations in biology where number-theoretic constants or sequences arise?

Fermat, before Wiles

On June 23, 1993, Princeton mathematician Andrew Wiles stunned the world with an announcement which eventually led to a proof, joint with Richard Taylor, of Fermat's Last Theorem: the equation xⁿ + yⁿ = zⁿ has no solutions in positive integers x, y, z, n, with n > 2. Prior to this, however, lots of exponents n had been ruled out from admitting any solutions. In fact, computer calculations (J. P. Buhler et al., Math. Comp. 61, 1993) had ruled out all exponents n <= 4,000,000. Describe some of this story: the partial successes, as well as the many false claims to have come up with complete solutions, from cranks and credible mathematicians alike. Focusing on a few small cases (e.g., n = 3) might illustrate some interesting number-theoretic techniques.

To Leap or not to Leap: The Calendar and the Irrational Number 365.242...

How often should February 29 come around? The history of the calendar, throughout civilization, shows a remarkable parallel between the ability to measure the length of a solar year and the degree to which leap years allow the calendar to stay on track. To some extent, the increments in accuracy reflect the sequence of rational approximations to the (irrational) number of days in a year given by the continued fraction expansion. Describe these developments. Why does the rational approximation inherent in the present calendar system depart from the rational approximations coming from the continued fraction? How have the calendar systems in other cultures dealt with irrational numbers?

Prime Testing: They Keep Getting Bigger

With the widespread use of public-key cryptography, there is an ever-increasing demand for larger and larger primes. How do we produce such large primes? Do current primality testing algorithms allow some "false" primes (e.g., psuedoprimes) to slip through? What about the record-breaking primes: What sorts of techniques allow us to focus on numbers of a special form and identify primes with hundreds of thousands of digits?

The Story of a Diophantine Equation: Elkies' Solution of A⁴ + B⁴ + C⁴ = D⁴

Harvard mathematician Noam Elkies (Math. Comp. 51, 1988) cracked a centuries-old problem by finding a solution in integers, with each on the order of millions or tens of millions. How did he do this? Certainly not by exhaustion! What led to the long-standing belief that this equation would admit no solutions, and what led Elkies to find his solution?

Elliptic Curves Primer

Elliptic curves are defined by an equation of the form y² = x³ + ax + b. Explain how the solutions (x,y) to such an equation (together with an added "point at infinity") admit an associative, commutative composition law, that is, they form an abelian group. This has applications to public-key cryptography. Explain a bit of the theory. Also explain why, for applications, it is preferable to work over a finite field (e.g., Z/p) rather than, say, over the rational numbers.

Some Equations You Just Can't Solve: Gödel's Incompleteness Theorem and Diophantine Equations

Explain Gödel's Incompleteness Theorem, and how this result implies the existence of (families of) Diophantine equations, whose solvability can be neither proved nor disproved under the regime of the standard arithmetic axioms.