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 |
(This and several following problems are from the International Mathematics Tournaments of the Towns Conference held in Beloretsk, Russia, summer 1993.) Proposer: Professor A. A. Egorov Consider the following diophantine equation in x and y: x^2 + (x+1)^2 + ... + (x+n-1)^2 = y^2, (*) where n is a given positive integer. If (*) has infinitely many solutions, we say that n is infinitely good. (a) Prove that 2, 11, 24 and 26 are infinitely good. (b) Prove that there are infinitely many infinitely good positive integers. If (*) has at least one solution with x>0, we say that n is very good. (c) Prove that an infinitely good positive integer is very good. (d) Prove that a positive integer which is very good but not infinitely good cannot be even. (e) Prove that 49 is very good but not infinitely good. (f) Prove that there are infinitely many positive integers which are very good but not infinitely good. If (*) has at least one solution, we say that n is good. (g) Prove that 25 is good but not very good. (h) Prove that there are no other positive integers which are good but not very good. If (*) has no solutions, we say that n is bad. (i) Prove that 3, 4, 5, 6, 7, 8, 9 and 10 are bad. (j) Prove that there are infinitely many bad positive integers. (k) Devise an efficient algorithm which classifies a positive integer as infinitely good, very good but not infinitely good, good but not very good, or bad.
T.R | Title | User | Personal Name | Date | Lines |
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1837.1 | (i) ??? | UTRTSC::BUIJS | Wed Feb 16 1994 07:59 | 14 | |
>>> Consider the following diophantine equation in x and y: >>> x^2 + (x+1)^2 + ... + (x+n-1)^2 = y^2, (*) >>> >>> If (*) has no solutions, we say that n is bad. >>> (i) Prove that 3, 4, 5, 6, 7, 8, 9 and 10 are bad. does "diophantine" have a restriction on the values for x and y? (I'm dutch so some english words or mathematical terms are unknown to me) if not, take n = 4, x = -1/2 and y = 3 x = -5/2 and y = 3 and (*) holds Marjan | |||||
1837.2 | RUSURE::EDP | Always mount a scratch monkey. | Wed Feb 16 1994 11:54 | 11 | |
Re .1: Diophantine equations are intended to be solved in integers only. -- edp Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75 To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from ftp.funet.fi. For FTP access, mail "help" message to DECWRL::FTPmail or open Upsar::Gateways. | |||||
1837.3 | Proof, give or take some typos | IOSG::CARLIN | Dick Carlin IOSG, Reading, England | Fri Mar 04 1994 17:16 | 183 |
1837.4 | SOP | IOSG::CARLIN | Dick Carlin IOSG, Reading, England | Fri Apr 22 1994 17:40 | 93 |
1837.5 | > -> >= and <= -> < :-) | IOSG::CARLIN | Dick Carlin IOSG, Reading, England | Sun Apr 24 1994 21:20 | 13 |