20161103, 07:14  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
1320_{8} Posts 
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. 
20161103, 08:41  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6,143 Posts 
How do you define "perfect square"? Must it be integers only? Or can it also be fractions and complex numbers etc.?

20161103, 09:15  #3  
"Robert Gerbicz"
Oct 2005
Hungary
2^{2}·367 Posts 
Quote:
"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. 

20161103, 09:26  #4 
"Matthew Anderson"
Dec 2010
Oregon, USA
1011010000_{2} Posts 
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 
20161103, 09:30  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}·3^{2}·5 Posts 
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 
20161103, 11:29  #6  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Last fiddled with by science_man_88 on 20161103 at 11:35 

20161103, 14:35  #7  
Aug 2006
3^{2}·5·7·19 Posts 
Quote:
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. 

20161103, 14:42  #8  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2×3×13×137 Posts 
Quote:
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 20161103 at 14:45 

20161103, 14:53  #9  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
Last fiddled with by science_man_88 on 20161103 at 14:55 

20161103, 17:44  #10  
Aug 2006
3^{2}·5·7·19 Posts 
Quote:


20161104, 06:49  #11 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}·3^{2}·5 Posts 
Hi Mersenneforum,
Thank you for your replies. C.R.Greathouse, you seem to have figured it out. Good show. Regards, Matthew 
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