mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2016-11-03, 07:14   #1
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

10110100002 Posts
Talking a number challenge

Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.
MattcAnderson is offline   Reply With Quote
Old 2016-11-03, 08:41   #2
retina
Undefined
 
retina's Avatar
 
"The unspeakable one"
Jun 2006
My evil lair

614310 Posts
Default

How do you define "perfect square"? Must it be integers only? Or can it also be fractions and complex numbers etc.?
retina is offline   Reply With Quote
Old 2016-11-03, 09:15   #3
R. Gerbicz
 
R. Gerbicz's Avatar
 
"Robert Gerbicz"
Oct 2005
Hungary

22·367 Posts
Default

Quote:
Originally Posted by MattcAnderson View Post
Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.
The original problem:
"Call a positive integer n good if the sum of n consecutive integers could be a perfect square, and bad otherwise. For example, 3 is good because 2+3+4=9=3^2. In Square sums, you were asked to find all bad numbers."

It is a quite different problem from the above, and has got a better wording. The problem is very well known.
R. Gerbicz is offline   Reply With Quote
Old 2016-11-03, 09:26   #4
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

10110100002 Posts
Default

Hi all,

@retina I should have posted that we want
to assume that n is an integer.

I did not want to consider fractions, irrationals, and other real numbers.

Further, I want to restrict this puzzle to the real numbers.

Complex numbers are out


Also, this problem is well known by those that well know it.
I copied it from a local University "POW" Problem Of the Week.
Luckily, I am still on their email distribution list.

Regards,
Matt
MattcAnderson is offline   Reply With Quote
Old 2016-11-03, 09:30   #5
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

24·32·5 Posts
Default

Hi mersenneforum

To be clear, perfect square numbers are
numbers like
0, 1, 4, 9, ...

I guess that was a definition by example

Regards
Matthew
MattcAnderson is offline   Reply With Quote
Old 2016-11-03, 11:29   #6
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

100000110000002 Posts
Default

Quote:
Originally Posted by MattcAnderson View Post
Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.
my guess basically most of them because x^2 is the sum of x numbers that average to x, so 1^2 = 1, 2^2 = 1+3, 3^2 = 2+3+4,4^2 =3+4+4+5. edit: the squares are known to be the sum of the first x odd integers as well. 1^2=1;2^2=1+3;3^2 = 1+3+5; etc.

Last fiddled with by science_man_88 on 2016-11-03 at 11:35
science_man_88 is offline   Reply With Quote
Old 2016-11-03, 14:35   #7
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

176116 Posts
Default

Quote:
Originally Posted by MattcAnderson View Post
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.
I'll follow Math Horizons and call a number n "good" if there is a sum of n consecutive integers which is square.

1 is good because 1 is a square. 2 is good because 4+5 = 3^2. 3 is good because 2 + 3 + 4 = 3^2. 4 is bad because n + n+1 + n+2 + n+3 = 4n + 6 is never a square. 5 is good because 3 + 4 + 5 + 6 + 7 = 5^2. So I get a very different list from you: 4, 12, 16, 20, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, ... which is A108269 in the OEIS.
CRGreathouse is offline   Reply With Quote
Old 2016-11-03, 14:42   #8
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

5·2,137 Posts
Default

Quote:
Originally Posted by CRGreathouse View Post
I'll follow Math Horizons and call a number n "good" if there is a sum of n consecutive integers which is square.
I believe that the OP meant:

Find solutions (m,n) in integers to the Diophantine equation m^2 = n(n+1)/2.

He further asserts that the the sequence of values for m is not in the OEIS.

Last fiddled with by xilman on 2016-11-03 at 14:45
xilman is online now   Reply With Quote
Old 2016-11-03, 14:53   #9
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

20C016 Posts
Default

Quote:
Originally Posted by xilman View Post
I believe that the OP meant:

Find solutions (m,n) in integers to the Diophantine equation m^2 = n(n+1)/2.

He further asserts that the the sequence of values for m is not in the OEIS.
a quick search with PARI shows those values he listed are a incomplete list of the n values actually. edit: with a more complete list of values you get https://oeis.org/A001108

Last fiddled with by science_man_88 on 2016-11-03 at 14:55
science_man_88 is offline   Reply With Quote
Old 2016-11-03, 17:44   #10
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

32·5·7·19 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
a quick search with PARI shows those values he listed are a incomplete list of the n values actually. edit: with a more complete list of values you get https://oeis.org/A001108
Right, or https://oeis.org/A001109 in the opposite direction.
CRGreathouse is offline   Reply With Quote
Old 2016-11-04, 06:49   #11
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

24×32×5 Posts
Default

Hi Mersenneforum,
Thank you for your replies. C.R.Greathouse, you seem to have figured it out. Good show.
Regards,
Matthew
MattcAnderson is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
When I was your age.....CHALLENGE petrw1 Lounge 14 2009-11-23 02:18
Challenge science_man_88 Miscellaneous Math 229 2009-09-07 08:08
rsa-640 challenge ValerieVonck Factoring 58 2005-10-24 15:54
Another challenge R.D. Silverman Programming 24 2005-07-27 21:08
Who is Challenge? JuanTutors PrimeNet 2 2004-07-22 12:56

All times are UTC. The time now is 14:38.

Sun May 16 14:38:25 UTC 2021 up 38 days, 9:19, 0 users, load averages: 4.51, 4.42, 4.15

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.