A Proof that Shoots its own Foot

Recently I began to get interested in odd perfect numbers. A perfect number is one that is equal to the sum of the divisors less than it. For example, 28 is divisible by 1, 2, 3, 4, 7, and 14 and these numbers sum to 28, so 28 is a perfect number. Euler proved that a number that is of the form 2ⁿ*(2ⁿ⁺¹-1) is perfect, and that all even perfect numbers are produced from this formula for some n. It is not known if an odd number can be perfect; it would have to satisfy so many conditions that it would make a lawyer's head swim. But no one has proved that there isn't one.

So it surprised me when I found one on the Internet! The number is 198,585,576,189. This number's factorization is 3²*7²*11²*13²*22,021. It can easily be shown that an odd perfect number has to be an odd prime times a square number. This number certainly has this form, provided of course that 22,021 is prime. It is not readily obvious that 22,021 is prime; it looks like it may be. The interesting thing about it is that if you assume that 22,021 is prime, then you can show that 198,585,576,189 is indeed a perfect number. But when doing this, you do something else that undermines this proof altogether. The proof that 198,585,576,189 is perfect also implies that 22,021 is not prime! In fact, it shows that 22,021 = 19²*61.

So here we have a case of proving the statement A -> B. But in so doing you also prove A -> ~A; i.e., that A implies not A, the statement that A is false. This destroys the proof altogether, because although it may be true that A -> B, that does not tell us anything, for if A -> ~A, then ~A must be true, and so A is false, so that A -> B is true but only because a false statement implies anything. It is the proof that shoots its own foot.

By the way, the page which shows the number 198,585,576,189 and shows that if 22,021 is assumed to be prime, then the number is perfect, is http://www.primepuzzles.net/puzzles/puzz_111.htm.

Redesign of Blogtrek

I decided to put some pizzazz to my Blogtrek page. I changed the colors of the sidebar and the top, and put a dark blue border around the entire page. This made some black text unreadable, so I changed it to white. Lastly, I designed a graphic meant to suggest infinity, in accord with my personal philosophy of "Just add one." I added in the phrase "...blogging where no one has blogged before...", in accordance with the similarity between Blogtrek and Startrek. Finally, I made the text area white, for maximum readability. The result is a nicer looking page, in my opinion.

Columbia Tragedy: Memorable or Full?

I rate news stories on their effect on the media. I have six ratings: Memorable, Full, Major, Front-Page, News, and Non-News. To qualify as full, the story must cover the entire newscast at night (hence the name full) and must cover half of the front page or more with a headline, but still leave room for the story. A Memorable event is noted by people asking each other in the future, "When were you when ___ occurred?" about the event. It will also fill the entire front page of a newspaper with a headline and will wipe out all other TV programming for at least 24 hours. Planeattack was certainly a Memorable Event. These are rare; so rare that only five have occurred in my lifetime.

So is the Columbia tragedy full or Memorable? It filled much of the front page of the newspaper but it left room for story. It dominated all the networks for times on 2003 Feb 1, but it left room for other programming later in the day. It did fill the entire news program on the 1st, so it is at least full. But is it memorable? I don't think so. Already other programming is coming back and the news coverage was only a few pages in the newspaper. So it rates as a high full event, somewhere between full and Memorable, but not quite memorable. It is about like the Challenger disaster was. But this is an event that we will remember for a long time, probably in the top 10 in my lifetime. I just hope that it does not put an end to space exploration.

The Columbia Disaster

I watched with horror the asteroid-like streaks across the sky on TV that was the remains of Columbia, which catastrophically broke up over the skies of Texas yesterday. It gave me the "not again" feeling, and reminded me of Challenger in 1986. I grieve for those people who went out into space, risking their lives, to extend the depth of human knowledge. The question is whether this is the end of the Space Shuttle or of space exploration. I don't think space exploration should ever end. It may be the one thing that saves our species, and it helps to extend the breadth of human knowledge; look at all we found out about the universe through the Hubble Telescope. But the Shuttle may have seen its last days. There should be one more Shuttle flight - to get the astronauts in the Space Station down. Then it should be considered as to whether we should continue flying this 1970s-technology craft again. A suggestion has been made to do as much through unmanned probes and robots as possible, and to send humans up to space in a small space plane when they are needed. The technology is certainly available to fly an airplane, have it turn into a rocket and fly into space. I think it should be worth considering. In any case, the Shuttle is full of old technology, and we need to get something that is up to date, not premillennium.

Heritage without Hate?

I saw a bumper sticker yesterday when shopping. It read "Heritage without Hate" and it had a Confederate battle flag on it. By this the author of the sticker presumably meant that he intended to celebrate and revere his Confederate ancestry; this is where he came from. He feels the flag symbolizes the heroism that Confederate soldiers displayed during the US Civil War. The problem is that he does not get the message across. I can see his point to some extent; after all, one of my ancestors fought on the Confederate side. But what means one thing to one person means another to another. Many African Americans feel that it does mean hate, even when the originator does not intend that. I feel that such a celebration of Confederate ancestry, including my own, needs to be done in a private manner, and that other heritages are more worthy of celebration. It is like Germans celebrating their heritage by displaying a swastika.

There is even a dispute in Volgograd, Russia, about the name of the place. It was called Stalingrad in one of the most stupendous battles in history, the Battle of Stalingrad in World War II. Many Russians want to change it back to Stalingrad because that is what their ancestors fought for. But like with the Confederate flag, I think this should be done in a more subdued fashion, for Stalin was one of the most brutal men in history. The town should remain Volgograd, naming it after a river, representing our planet Earth - a heritage certainly worth supporting.