Sunday, August 05, 2007
A Mathematical Puzzle
The French mathematician Fermat claimed to have discovered a proof that the equation: A^n + B^n = C^n,
[where "^n" means "raised to the power of n"]
has integer solutions only for n=2. It is easy to show that the numerical solutions for A, B, C are (3,4,5), (5,12,13), (7,24,25), and so on. More than 200 years after Fermat wrote the tantalizing note in the margin of a book some mathematician was able to find a proof.
I have been looking at the equation A^n + B^n + C^n = D^n where n=3 and A, B, C, and D are integers. I haven't been able to develop an expression for generating the solutions. I have found a few solutions by trial and error for A, B, C, and D. As in the case of the Fermat equation, I have looked only at cases D = C+1. Here are a few solutions:
(3,4,5,6), (1,6,8,9), (3,10,18,19), (2,17,40,41), (14,23,70,71), (12,31,102,103)
I expect that some mathematical wizard who happens to read this blog will COMMENT that this problem has already been examined extensively and that I'm just wasting my time to repeat work that has been done already. I will welcome the comment. Otherwise, I may continue plodding on and finding more solutions to the problem without hitting on a general formula for generating solutions.
In the case of the Fermat equation, for any odd number A, B = (A^2-1)/2.
Friday, August 03, 2007
Bush's threat to veto the SCHIP reauthorization bill
- He is a determined ideologue who would rather placate the extreme "small government" wing of the Republican Party than the more moderate majority of Republicans and Independents, to say nothing of Democrats. His guiding philosopher in this endeavor is Grover Norquist, not Jesus Christ.
- He shows how hollow the slogan "compassionate conservative" is. Telling children of lower middle class parents that they can have all the health care their parents are able and willing to pay for is certainly "conservative." It is not compassionate, at least not toward the parents and children who are affected.
- He argues that enlarging the SCHIP program is a step toward a publicly funded universal health care system and must be stopped because it leads to socialism. (Horrors!)
- He has compassion for the plight of the poor tobacco companies whose products would be subjected to additional taxation to pay for the additional children.
- He has compassion for the HMO's and insurance companies who might lose clients if the SCHIP program is enlarged. Parents might abandon their own unsatisfactory HMO or health insurance plan and enrol instead in SCHIP.
There is a lesson for us who voted hopefully for Democrats last year. Even with a Democratic majority in both houses of Congress, we are not going to see any progress toward providing access to good health care to those who don't already have it as long as George W. Bush is President.
Thursday, August 02, 2007
Rights and Rights
“Clint Bolick is one of America’s greatest champions of freedom. In this accessible and timely book, he challenges the political left and right to set aside time-worn and ill-defined complaints of “judicial activism.” His writing is a call to principled reciprocity: those who readily embrace speech and associational rights ought not begrudge like recognition of property rights or our innate desire to pursue a lawful occupation. At a time when Supreme Court Justices are mounting national platforms to defend judicial independence, Bolick reminds those privileged to serve in the third branch that judges are given independence for a reason—and it is neither unthinking deference to legislative majority nor the imposition of personal will.”
What intrigued me about the review was the request that those of us who value freedom of speech should "not begrudge like recognition of property rights...." I know that some of us who treasure the First Amendment right to freedom of speech tend to forget that there are limits on freedom of speech. The right is not absolute. The classic example is that one does not have the right to shout "fire" in a crowded theater. Professor Kmiec is a conservative but also a reasonable man. I am sure that he recognizes that a property right or a right of intellectual property is also not absolute. I own my house; my wife and I paid off the mortgage thirteen years ago. However, I do not have the right to set up a slaughter house in my back yard, nor do I have the right to operate a jet aircraft engine at full power which generates a sound level of 160 decibels. My neighbors have the right not to be annoyed or harmed by what I choose to do with my property.
I wonder whether our new conservative Supreme Court will recognize sensible and logical limits on the use of provate property.