First a puzzle: How far can you count if you are only allowed to say numbers which can be written as a formula using the digits 1,2,3 and 4 once each, and the usual maths operations?
This was a puzzle someone posted to rec.puzzles. I have put together a page with my best answer
.My interest in number theory began when I was at high school. I was always drawn by the magic and history of prime numbers so I was thrilled to find that George Woltman had organised a project to find new large Mersenne Primes. So far I have just a few negative results but it is good to find a use for the full power of my Pentium PC. The project has already had its first successes in finding the 35th, 36th and 37th known Mersenne primes and is continuing to search for even bigger one. Perhaps the first million digit prime will come from this effort.
For me the true beauty of maths is demonstrated when people discover unexpected connections between previously unrelated ideas. The canonical example of this is Euler's simple formula connecting fundamental constants of geometry and analysis.
ei&pi = -1
A more recent and difficult example is the equivalence of eliptic curves and modular forms according to the Tinamen conjecture who's solution was the key to Wiles proof of Fermat's last theorem. The story behind Wiles discovery was told in a BBC Horizon program. A transcript is available on the web.
As a student I was impressed by the use of analytic functions to solve diophantine equations and other problems in discrete mathematics. I never followed the subject far enough to really understand it all but I like the following alternative proof of the infinitude of prime numbers which demonstrates some strange connections between to the prime number theorem, the zeta function and the transcendence of pi.
Suppose there were only a finite number of primes pi. Then consider the following number defined as a finite product
R = product of pi2/(pi2-1)
By a simple series expansion we get,
x/(x-1) = 1 + 1/x + 1/x2 + 1/x3 + ...
Substitute this into the defintion of R and examine its expansion into an infinite series of terms. Each term is the reciprocal of a product of powers of primes and by the prime number theorem on the uniqueness of prime factorisation the sum simplifies to,
R = 1 + 1/22 + 1/32 + 1/42 + ...
Which by definition of the zeta function gives
R = zeta(2) = pi2/6
By the original hypotheses that there are a finite number of primes R must be a rational number, yet pi is transcendental, reductio ad absurdum...
Admittedly I glossed over a couple of steps in the proof. The value of zeta(2) can be found in various ways using fourier analysis for example. The proof that pi is transcendental is, of course, much more difficult than Euclid's original proof of the infinitude of primes. In fact the final solution to that problem came many centuries later and showed that there was no way to square the circle using compass and ruler, a problem which Euclid was far from solving. This is the hidden joke in my proof.
Now that Fermat's Last Theorem has been proved many people are asking what is it's successor. In response Kevin Brown has put together an interesting page of unsolved mathematical problems called the Most Wanted List. This includes an extensive collection of number theory problems. Among them is an old problem which has always interested me. It concerns the 4-tuple of numbers 1, 3, 8, 120 and others like it. They have the property that the product of any two of the four numbers is one less than a square.
1*3 = 22-1 1*120 = 112-1 1*8 = 32-1 3*120 = 192-1 3*8 = 52-1 8*120 = 312-1
You can read Kevin Brown's summary of a discussion I started about this in sci.math a couple of years back. In 1978 I had heard about this problem in the mathematical puzzles section of the Bulletin of the British Computer Society. They asked if there is a fifth number which can be added to the sequence 1, 3, 8, 120 which gives one less than a square when multiplied by any one of them.
In sci.math Richard Pinch reminded me that Baker and Davenport had solved this problem in 1969. Richard has a computer program which can quickly determine whther or not there is a fifth number corresponding to any given 4-tuple.
No sequence of 5 such positive integers is known but there are many sets of 4. I have conjectured that they are all solutions of this diophantine equation:
(a + b - c - d)2 = 4(ab+1)(cd+1)
The converse can be proven. I.e all solutions in positive integers of this diophantine equation have the property that the product of any two is one less than a square. This I proved by infinite descent.
I believe there is still much to be learned from studying this problem and I have written a longer article on the subject.
My Erdos number is less than or equal to 6 as follows:
Philip Gibbs
Ian Barbour
John Kogut
James Bjorken
Sheldon Glashow
Daniel Kleitman
Paul Erdos
This page was last updated 13 February 1999.
