NET TOPICS

1..Prime numbers have intrigued mathematicians for centuries. It is an
important sub-branch of mathematics called "number theory". Despite the
efforts of the best mathematical minds over the centuries, there is no
general "formula" for generating prime numbers. There are some
approximations, and there are theorems predicting the number of prime
numbers less than a particular upper bound. As of 2003 the largest known
prime number is:
The 39th Mersenne prime


m39 = 213,466,917-1

is the largest known prime number (as of the time of this writing, May 23,
2003). In the decimal system it requires 4,053,946 digits to be written
fully. Most of this page is taken up by those digits.
2..The system of quaternions is an extension of the complex number system in
the same way that the complex number system is an extension of the real
number system. Recall that you treat 2 + 3i algebraically just as you would
treat
2 + 3x except that when multiplying you replace any occurrence of i*i by -1.
You would treat a quaternion such as 2 + 3i + 5j - 2k algebraically just as
you would treat 2 + 3x + 5y - 2z except that you replace products of i, j,
and k according to the following rules:
i * i = -1, j * j = -1, k * k = -1, i * j = k, j * k = i, k * i = j,
j * i = -k, k * j = -i, and i * k = -j.
As you observed, this does make multiplication not commutative
, but is not any weirder than the i * i = -1 in the complex
numbers or the non-commutativity of matrix multiplication
3..If K is a prime and M(K) = 2^K-1 is also a prime (
called a Mersenne
prime) then P(K) =2^(K-1)*M(K) is a perfect number (the sum of all of its
proper divisors is equal to P(K)).
4.Christian Goldbach (1690-1764), in a letter to Euler dated February, 16,
1745, stated that every even number equal to or greater than 6 is the sum of
two odd primes in one or more ways. The first sentence is a direct quote
from THE LORE OF PRIME NUMBERS by G. P. Loweke, p 68. The conjecture
remains unsettled, but computers have verified the conclusion for all even N
below very large bounds (Loweke noted the bound of 100,000 but that is quite
old, I believe). A related conjecture is that every even number is a
difference of two primes in infinitely many ways! Loweke ascribes this
statement to A. de Polignac around 1849.
5.. Fermat's Last Theorem states that the equation
a^n + b^n = c^n
has no integer solutions for n > 2. The theorem can be easily proved for
some specific values of _n_, for example, n = 3. The hard part is proving
it in general.
6..
Two positive integers are "relatively prime" if their prime factorizations
have no common prime factors. For example, if you "reduce a rational
fraction to its lowest terms"
then the numerator and denominator must be relatively prime.
7..The ancient greeks decided (probably based on experiments: try using a cord
to estimate the circumference of some cylinders) that the perimeter of a
circle (and other symmetric objects) must be a constant multiple of its
diameter. A little later attempts were made to estimate the value of the
constant using the geometry of inscribed polygons. Much later, when
symbolism became more popular in mathematics, "pi" (the first letter in the
greek word for perimeter) started to gain acceptance for representing this
famous constant. Full details on how "they" get it are too long to type
here. Perhaps the earliest estimate for PI was 3. I believe that Archime-
des gets credit for using geometry to improve the estimate to 22/7. The
greeks at that time loved pure fractions and used nothing like our many
decimal representations.