In studying whole
numbers, mathematicians have
discovered a variety of surprising
patterns. One of the most important results of elementary
number theory is the so-called law of quadratic reciprocity, which
links prime numbers (those evenly divisible only by themselves
and one) and perfect squares
(whole numbers multiplied by themselves). [Such
linkage is a common feature of Brunardot's Theorem, which links Natural
Prime numbers with every integer. -B]
For a positive integer, d, the law describes the primes, p,
for which there exists a number x such that dividing the square
of x by p gives the same remainder as dividing d by
p. For example, if p is 23 and d is 3,
there's a solution when x is 7. Dividing 72, or
49 by 23 leaves the remainder 3, as does dividing 3 by 23.
The law specifies the relationship that must hold between p and d
for an x to exist. [Brunardot's
Theorem goes far beyond this relationship to much more complex
situations. See:
Brunardot's Harmonic Ellipses.
-B]
Mathematicians have sought to
generalize the law to cover cases such as when
the numbers are not squares but cubes or larger powers. [Brunardot's
Theorem holds for every integer. -B] In the
late 1960s, Robert Langlands, now at the Institute for Advanced Study in
Princeton, N.J., describes how such
reciprocity laws might work in general context in number theory, in
which integers represent a special case.
[This is what Brunardot's Theorem does . . . and much more. -B]
Michael Harris of the
University of Paris VII and Richard
Taylor of Harvard University recently
proved this conjecture, called the "local Langlands
correspondence." It was also
independently established as a theorem shortly afterward by Guy
Henniart of the University of Paris-South.
[Brunardot published a much more general and extensive theory in the
Spring of 1994. -B]
The proof of the theorem "in full generality
represents a milestone
[Harris, et al.'s theory is not near "full
generality." -B] in algebraic number theory," [Brunardot's
Theorem unifies the fundamental phenomena of physics and . . . Science,
Theology, and Philosophy. -B] mathematician Jonathan Rogawski of the University of California, Los
Angeles remarks in the January Notices
of the American Mathematical Society. |
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