[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

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

1837.0. "Beloretsk 1" by RUSURE::EDP (Always mount a scratch monkey.) Tue Feb 01 1994 16:09

    (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.RTitleUserPersonal
Name
DateLines
1837.1(i) ???UTRTSC::BUIJSWed Feb 16 1994 07:5914
>>>    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
1837.2RUSURE::EDPAlways mount a scratch monkey.Wed Feb 16 1994 11:5411
    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.
1837.3Proof, give or take some typosIOSG::CARLINDick Carlin IOSG, Reading, EnglandFri Mar 04 1994 17:16183
1837.4SOPIOSG::CARLINDick Carlin IOSG, Reading, EnglandFri Apr 22 1994 17:4093
1837.5> -> >= and <= -> < :-)IOSG::CARLINDick Carlin IOSG, Reading, EnglandSun Apr 24 1994 21:2013