| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
As shown in many number theory books, given any positive integer whose square root is irrational, the root can be represented as a repeating continued fraction. For example: sqrt(983) = 31 2 1 5 31 5 1 2 62 2 1 5 31 5 1 2 62 2 1 5 31 5 1 2 62 ... In this case, the "2 1 5 31 5 1 2 62" repeats forever. In case you're not familiar with continued fractions, this means sqrt(983) = 31+1/(2+1/(1+1/(5+1/(31+1/(5+1/(1+1/(2+1/... My conjecture: The pattern of numbers not counting the last number of the pattern is always a palindrome. In our example, it's easy to see we have a palindrome: 2 1 5 31 5 1 2 Am I the first to recognize this fact ? Is it interesting or just ho hum ? I'm not sure how to go about proving it yet. /Eric
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1353.1 | intersting.. | SMAUG::ABBASI | Tue Dec 11 1990 15:22 | 5 | |
Have you tested OSMAN conjecture on a large number of such positive
integers whose square root is irrational? may be by writing a program?
your conjecture is intersting if it comes to be true..
/nasser
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| 1353.2 | well spotted | HERON::BUCHANAN | combinatorial bomb squad | Wed Dec 12 1990 04:54 | 5 |
It's true. I have the proof somewhere. Galois did it when he was a brat. Regards, Andrew. | |||||
| 1353.3 | But then, | VMSDEV::HALLYB | The Smart Money was on Goliath | Wed Dec 12 1990 08:51 | 1 |
Galois DIED when he was still a brat! | |||||
| 1353.4 | poor chap | HERON::BUCHANAN | combinatorial bomb squad | Wed Dec 12 1990 09:22 | 12 |
> Galois DIED when he was still a brat! 1829 [the year the first railway opened, from Stockton to Darlington in England] saw the publication of Galois' first paper, on continued fractions. He was aged 18. He died 31 May 1832, of peritonitis following an absurd duel the previous day. Fortunately for mathematics, he had taken the precaution the evening before the duel to write a letter to his friend Auguste Chevalier outlining the connection between groups and polynomial equations, including his fundamental solubility theorem, and many other ideas. Regards, Andrew. | |||||
| 1353.5 | Leads to Solution of Pell's Equation | TROA09::RITCHE | From the desk of Allen Ritche... | Wed Dec 12 1990 15:57 | 18 |
Yes, it's true and a very interesting and useful fact.
I have an old paperback at home by C.D. Olds entitled "Continued
Fractions" (the book was part of a mathematics series published
in the 60's I think).
It's easy reading and contains a proof of .0
It also develops the theory of c.f.'s from scratch leading nicely into
applications; i.e. solving Diophantine equations of the form
ax + by = c using finite c.f's; then to infinite c.f.'s of quadratic
surds and in particular how to solve Pell's Equation using the
c.f. periodic expansion of a pure quadradic, sqrt(N).
What's really interesting is the result that .0 helps solve in integers
Pell's Equation x� - Ny� = 1
Allen
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