 mersenneforum.org a number challenge
 Register FAQ Search Today's Posts Mark Forums Read  2016-11-03, 07:14 #1 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 13208 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.   2016-11-03, 08:41 #2 retina 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.?   2016-11-03, 09:15   #3
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

22·367 Posts Quote:
 Originally Posted by MattcAnderson 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.   2016-11-03, 09:26 #4 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 10110100002 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   2016-11-03, 09:30 #5 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 24·32·5 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   2016-11-03, 11:29   #6
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts Quote:
 Originally Posted by MattcAnderson 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   2016-11-03, 14:35   #7
CRGreathouse

Aug 2006

32·5·7·19 Posts Quote:
 Originally Posted by MattcAnderson 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.   2016-11-03, 14:42   #8
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

2×3×13×137 Posts Quote:
 Originally Posted by CRGreathouse 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   2016-11-03, 14:53   #9
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts Quote:
 Originally Posted by xilman 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   2016-11-03, 17:44   #10
CRGreathouse

Aug 2006

32·5·7·19 Posts Quote:
 Originally Posted by science_man_88 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.   2016-11-04, 06:49 #11 MattcAnderson   "Matthew Anderson" Dec 2010 Oregon, USA 24·32·5 Posts Hi Mersenneforum, Thank you for your replies. C.R.Greathouse, you seem to have figured it out. Good show. Regards, Matthew   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post petrw1 Lounge 14 2009-11-23 02:18 science_man_88 Miscellaneous Math 229 2009-09-07 08:08 ValerieVonck Factoring 58 2005-10-24 15:54 R.D. Silverman Programming 24 2005-07-27 21:08 JuanTutors PrimeNet 2 2004-07-22 12:56

All times are UTC. The time now is 16:18.

Sun May 16 16:18:46 UTC 2021 up 38 days, 10:59, 1 user, load averages: 3.00, 2.14, 2.11