Taming a Beast called UNIX

The year was 1988. Voices hushed in trepidation, the apprentices gathered in a dingy corner of the Engineering building. In the undergraduate computer lab we assembled to offer sacrifices to the mainframe, hunched over the fading amber or green glow of hesitant dumb terminals, all connected to the mighty UNIX (VAX/VMS) mainframe. Poor first year […]

Fermat’s Last Theorem

Fermat’s Last Theorem states that:

There are no positive integers x, y, and z  with x^n + y^n = z^n if n is an integer greater than 2.

For n = 2 there are many solutions:

( 3, 4, 5) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)
( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) … the Pythagorean triples.

In the margin of his copy of the Arithmetica of Diophantus the French jurist Fermat wrote circa 1637 that for greater n no such triples can be found; he added that he had a marvelous proof for this, which however the margin was too small to contain:

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet.

Some of the greatest mathematical minds in history have applied their genius to this problem, none succeeded. Every other result which Fermat had announced in like manner had long ago been dealt with; only this one, the last, remained. Finally, 358 years after Fermat wrote his theorem, Andrew Wiles saw an application of elliptic curves and modular forms (20th century mathematical constructs, not known to earlier mathematicians) that could contribute to the proof.


Let n, a, b, c, \in \mathbb{Z} with n > 2.
If a^n + b^n  = c^n then abc  = 0.


The proof follows a program formulated around 1985 by Frey and Serre[F,S]. By classical results of Fermat, Euler, Dirichlet, Legendre, and Lamé, we may assume that n = p, an odd prime ≥ 11. Suppose a, b, c  \in \mathbb{Z} , abc \neq 0, and a^p + b^p  = c^p. Without loss of generality we may assume that 2|a and b \equiv 1 \bmod 4. Frey [F] observed that the elliptic curve E : y^2 = x(x -a^p )(x + b^p) has the following ‘remarkable’ properties:
(1) E is semistable with conductor N_E = \prod_{{l}|abc} {l} ; and
(2) \overline{\rho}_{E,p} is unramified outside 2p and is flat at p .
By the modularity theorem of Wiles and Taylor-Wiles [W,T-W], there is an eigenform f \in S_2 (\Gamma_0(N_E)) such that \rho_{f,p} = \rho_{E,p}. A theorem of Mazur implies that E,p is irreducible, so Ribet’s theorem [R] produces a Hecke eigenform g \in S_2 (\Gamma_0(2)) such that \rho_{g,p} \equiv \rho_{f,p} \bmod {\tt p} for some {\tt p}|p. But X_0(2) has genus zero, so S_2 (\Gamma_0(2)) = 0 . This is a contradiction and Fermat’s Last Theorem follows. Q.E.D.

References: Continue reading “Fermat’s Last Theorem”

RE: Darwin & The Deity

Faith and Theology: Ten propositions on Darwin and the deity Should it be of concern to Christians that Darwin was never more than a nominal believer? – well, are you concerned whether your surgeon, mechanic, or hair stylist goes to church? Of course not. Your only concern is that she wields a scalpel, wrench, or […]

USA is going crazy.

Rick Santelli vented a forceful and furious rant against Obama’s multi-trillion stimulus package — many people suspect it will only cause further distortions in a deeply troubled economy, and the only realistic solution is to let people suffer the consequences of vast overspending. Anything else is just an attempt to defy the law of gravity. Personally I don’t know — I hope America can reinvent itself, but there’s a real sense of panic emerging from the States right now.

Even if it was some kind of staged PR exercise, what Santelli actually said is entirely correct. I’ve been saving doggedly all these years, watching everyone jump on board the property gravy train, and there’s NO WAY that I would willingly subsidize them. As Peter Schiff said, the President should be defending the dollar, not embarking on mad inflationary schemes. Obama’s plan is fundamentally unjust.

I Bought an Expensive House. My Bad, Not Yours

The only people affected by plummeting real estate prices are the ones who bought a house that cost more than they could afford, hoping for a spike in value so they could sell at a profit or take out a new loan based on an increased value. Their home wasn’t just a place to live; it was an investment they thought they could liquefy at will. If we’re saving these poor souls from the 26.7% drop in their investment, we should give twice as much aid to everyone who has lost approximately 50% in the stock market since its peak.

Transcript over the jump.. Continue reading “USA is going crazy.”