mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2007-07-30, 19:51   #1
davar55
 
davar55's Avatar
 
May 2004
New York City

5×7×112 Posts
Default Sums of Triangular Numbers

Show that any non-negative integer can be written
as the sum of three (or fewer) triangular numbers.

(Fermat claimed he had proven this and more,
and Gauss proved it as a youth.)

For reference, the triangular numbers are just
T(n) = 1+2+3+...+n = n(n+1)/2, T(0) = 0.

(I don't have the solution, and welcome a
re-solution, a short proof, some insight,
or a good pointer.)
davar55 is offline   Reply With Quote
Old 2007-07-31, 00:54   #2
R. Gerbicz
 
R. Gerbicz's Avatar
 
"Robert Gerbicz"
Oct 2005
Hungary

157310 Posts
Default

Quote:
Originally Posted by davar55 View Post
Show that any non-negative integer can be written
as the sum of three (or fewer) triangular numbers.
So for every n>=0 integer you have to find a,b,c >=0 integers for that:
n=a*(a+1)/2+b*(b+1)/2+c*(c+1)/2
Multiple the equation by 8:
8*n=4*a*(a+1)+4*b*(b+1)+4*c*(c+1)
Add 3:
8*n+3=(2*a+1)^2+(2*b+1)^2+(2*c+1)^2
Let x=2*a+1,y=2*b+1,z=2*c+1 positive integers, so
8*n+3=x^2+y^2+z^2
But if this holds then x,y,z are odd numbers because s^2==0,1 mod 4 so x,y,z are odd numbers and using the equation the solution of the original equation is a=(x-1)/2
b=(y-1)/2
c=(z-1)/2

But Gauss proved that if k!=4^u*(8*w+7) ( for some non negative u,w integers) then k is the sum of at most 3 square numbers. This is a very tricky and hard proof, as I can remember about 13 pages long. From this you can get the proof because 8*n+3!=4^u*(8*w+7).
R. Gerbicz is offline   Reply With Quote
Old 2007-07-31, 02:09   #3
alpertron
 
alpertron's Avatar
 
Aug 2002
Buenos Aires, Argentina

1,453 Posts
Default

By the way you can find the decomposition in sum of three squares using my applet on: http://www.alpertron.com.ar/FSQUARES.HTM
alpertron is offline   Reply With Quote
Old 2007-08-03, 11:59   #4
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22×33×19 Posts
Cool Lagrange.

Quote:
Originally Posted by davar55 View Post
Show that any non-negative integer can be written
as the sum of three (or fewer) triangular numbers.

(Fermat claimed he had proven this and more,
and Gauss proved it as a youth.)

For reference, the triangular numbers are just
T(n) = 1+2+3+...+n = n(n+1)/2, T(0) = 0.

(I don't have the solution, and welcome a
re-solution, a short proof, some insight,
or a good pointer.)
:surprised

It is well known that two consecutive numbers form a square number.

However J.L. Lagrange, Italian French math'cian (1736-1813) proved that any natural number can be expressed as the sum of 4 square numbers Therefore I conclude that 2x4 triangular numbers are required to express any integer.

Mally

Last fiddled with by mfgoode on 2007-08-03 at 12:00
mfgoode is offline   Reply With Quote
Old 2007-08-03, 12:40   #5
axn
 
axn's Avatar
 
Jun 2003

124018 Posts
Default

Quote:
Originally Posted by mfgoode View Post
Therefore I conclude that 2x4 triangular numbers are required to express any integer.
Necessary or sufficient? If the former, it can be trivially shown wrong. If the latter, it doesn't answer the puzzle.

Last fiddled with by axn on 2007-08-03 at 12:40
axn is offline   Reply With Quote
Old 2007-08-03, 14:05   #6
R.D. Silverman
 
R.D. Silverman's Avatar
 
"Bob Silverman"
Nov 2003
North of Boston

746010 Posts
Default

Quote:
Originally Posted by mfgoode View Post
:surprised

It is well known that two consecutive numbers form a square number.




Gibberish. This is not a statement of mathematics. "form a square number"
is NONSENSE. Explain how 5 & 6 "form a square number".

And you have the ridiculous arrogance to submit a paper for publication????
You can't even write a single, cogent, mathematical statement.
R.D. Silverman is offline   Reply With Quote
Old 2007-08-03, 17:02   #7
xilman
Bamboozled!
 
xilman's Avatar
 
"๐’‰บ๐’ŒŒ๐’‡ท๐’†ท๐’€ญ"
May 2003
Down not across

32·5·11·23 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post




Gibberish. This is not a statement of mathematics. "form a square number"
is NONSENSE. Explain how 5 & 6 "form a square number".

And you have the ridiculous arrogance to submit a paper for publication????
You can't even write a single, cogent, mathematical statement.
It's easy to see that Mally omitted the word "triangular" between "consecutive" and "numbers" and that such an omission is the sort of typo that most of us make all too often.

However, using "form" instead of "sum to" is remarkably sloppy nomenclature.

Paul

Last fiddled with by xilman on 2007-08-03 at 17:03 Reason: Fixed typo of my own!
xilman is online now   Reply With Quote
Old 2007-08-03, 19:34   #8
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22×33×19 Posts
Post Sloppy nomenclature!

Quote:
Originally Posted by xilman View Post
It's easy to see that Mally omitted the word "triangular" between "consecutive" and "numbers" and that such an omission is the sort of typo that most of us make all too often.

However, using "form" instead of "sum to" is remarkably sloppy nomenclature.

Paul


Thank you Paul for the back up of my typo.

I copy from notes I make up in short form ( yes form not sum!) of the various posts. If you check up the timing on the number of posts I wrote they were four, one after another of the thread, " My maths paper" This one was by far the last before I logged off. So the eyes do get tired to check them carefully esp. in mathematics.

And this after my p.c. giving a lot of trouble, my cell phone packing up and listening to the comments of my wife on the current program she is watching!

I admit I used the sloppy word 'form' to 'sum' and it is not correct in maths of today. Or is it so? well lets see.

I quote from Ore in his excellent book for beginners in number theory 'Invitation to number theory' esp. compiled for the 'New mathematical Library' by the School Mathematics Group, for the introduction of math theory in its various disciplines to high school students.

QUOTE" The geometric manner of expression is one of our many legacies from Greek mathematical thought. The Greeks preferred to think of numbers, including the integers, as geometric quantities." Hence a square is 'formed' by an array of numbers in a square of equal sides A rectangale by unequal sides.

And what are these but shapes and form, viisible to the naked eye? I am not condoning my sloppiness but I present this point of view.

I would like to ask Dr. Silverman how do you define a square as different from a rectangle and why some numbers are called rectangular and the other square?

Quote: "Some numbers cannot be represented as rectangular numbers except in the trivial way that one strings along the points in a single row; for instance 5 can be represented as a rectangular number only by taking one side to be 1 and the other to be 5" Hence 5 is called a prime.

What is the actual geometrical meaning of a prime number ? Well thats it above.

And why was 1 not, and is not a prime number? Why ?

The arithmetical concept of number was derived from geometry so how about a good geometrical explanation before we can ever conceive of a number?

I am thankful to both Axn1 and Mr. Silverman for pointing out the glaring error in my post but what has been left out, whch really matters, is am I right or wrong? If wrong then please point out a different interpretation of the thread.

Mally

Last fiddled with by mfgoode on 2007-08-03 at 19:41
mfgoode is offline   Reply With Quote
Old 2007-08-03, 20:01   #9
wblipp
 
wblipp's Avatar
 
"William"
May 2003
New Haven

2,371 Posts
Default

Quote:
Originally Posted by mfgoode View Post
whch really matters, is am I right or wrong?
I guess you mean:

Quote:
Originally Posted by mfgoode View Post
I conclude that 2x4 triangular numbers are required to express any integer.
You are wrong. Fermat's polygonal number theorem is that three triangle numbers or four square numbers or 5 pentagonal numbers ... or n n-gon numbers are sufficient.

http://mathworld.wolfram.com/Fermats...erTheorem.html

http://mathworld.wolfram.com/TriangularNumber.html
wblipp is offline   Reply With Quote
Old 2007-08-04, 06:50   #10
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22·33·19 Posts
Thumbs up A revelation.

Quote:
Originally Posted by wblipp View Post
I guess you mean:



You are wrong. Fermat's polygonal number theorem is that three triangle numbers or four square numbers or 5 pentagonal numbers ... or n n-gon numbers are sufficient.

http://mathworld.wolfram.com/Fermats...erTheorem.html

http://mathworld.wolfram.com/TriangularNumber.html


Thank you William once again for your brilliant and revealing reply.

Fermat was really, though a lawyer by profession. one of the greatest in number theory. It gives us all hope that we can rise above our professions to amazing heights no matter what it is.

I have never been aware of this theorem and the beauty and elegance it exhibits. Even if a blind man is told this, you know, three triangulars, 4 squares, 5 pentagonals and in general n, n-gons the sheer precise proportion and cadence like a musical chord in harmony, is enough to believe it. In short it makes sense.

Is'nt your straight forward reply 'You are wrong' and 'this is so' better than calling one all the names under the sun?

I find if one cannot enlighten then refrain from personal criticsm!

Thank you once again.

Mally
mfgoode is offline   Reply With Quote
Old 2007-08-04, 07:19   #11
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

80416 Posts
Lightbulb A sequel.



Talking about beauty and elegance in Mathematics the great British number theory specialist along with Little Wood was asked an evaluation on The great Indian Mathematician Ramanujan's papers.

I am writing ad lib and use my own words unless some brilliant spark in the forum directs me to an URL with the actual words of Hardy. It is the content in information that counts not its presentation.

Well Hardy saw the numerous papers on elliptic functions, partitions, etc. which Hardy could not understand so he replied in this vein. Of course some of them Hardy and Ram worked on, together to make them more complete.

"There is so much beauty in these results that it has to be right" or to that effect.

Mally
mfgoode is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Sequences and sums Microraptor Homework Help 10 2011-02-25 08:12
Sums of three squares CRGreathouse Math 6 2009-11-06 19:20
Prime Sums #3 davar55 Puzzles 2 2008-08-13 12:37
Prime Sums davar55 Puzzles 11 2008-03-31 05:24
Prime Sums #2 davar55 Puzzles 1 2008-03-19 14:12

All times are UTC. The time now is 12:02.


Tue Jul 5 12:02:40 UTC 2022 up 82 days, 10:03, 1 user, load averages: 1.74, 1.70, 1.50

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, 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.

โ‰  ยฑ โˆ“ รท ร— ยท โˆ’ โˆš โ€ฐ โŠ— โŠ• โŠ– โŠ˜ โŠ™ โ‰ค โ‰ฅ โ‰ฆ โ‰ง โ‰จ โ‰ฉ โ‰บ โ‰ป โ‰ผ โ‰ฝ โŠ โŠ โŠ‘ โŠ’ ยฒ ยณ ยฐ
โˆ  โˆŸ ยฐ โ‰… ~ โ€– โŸ‚ โซ›
โ‰ก โ‰œ โ‰ˆ โˆ โˆž โ‰ช โ‰ซ โŒŠโŒ‹ โŒˆโŒ‰ โˆ˜ โˆ โˆ โˆ‘ โˆง โˆจ โˆฉ โˆช โจ€ โŠ• โŠ— ๐–• ๐–– ๐–— โŠฒ โŠณ
โˆ… โˆ– โˆ โ†ฆ โ†ฃ โˆฉ โˆช โŠ† โŠ‚ โŠ„ โŠŠ โŠ‡ โŠƒ โŠ… โŠ‹ โŠ– โˆˆ โˆ‰ โˆ‹ โˆŒ โ„• โ„ค โ„š โ„ โ„‚ โ„ต โ„ถ โ„ท โ„ธ ๐“Ÿ
ยฌ โˆจ โˆง โŠ• โ†’ โ† โ‡’ โ‡ โ‡” โˆ€ โˆƒ โˆ„ โˆด โˆต โŠค โŠฅ โŠข โŠจ โซค โŠฃ โ€ฆ โ‹ฏ โ‹ฎ โ‹ฐ โ‹ฑ
โˆซ โˆฌ โˆญ โˆฎ โˆฏ โˆฐ โˆ‡ โˆ† ฮด โˆ‚ โ„ฑ โ„’ โ„“
๐›ข๐›ผ ๐›ฃ๐›ฝ ๐›ค๐›พ ๐›ฅ๐›ฟ ๐›ฆ๐œ€๐œ– ๐›ง๐œ ๐›จ๐œ‚ ๐›ฉ๐œƒ๐œ— ๐›ช๐œ„ ๐›ซ๐œ… ๐›ฌ๐œ† ๐›ญ๐œ‡ ๐›ฎ๐œˆ ๐›ฏ๐œ‰ ๐›ฐ๐œŠ ๐›ฑ๐œ‹ ๐›ฒ๐œŒ ๐›ด๐œŽ๐œ ๐›ต๐œ ๐›ถ๐œ ๐›ท๐œ™๐œ‘ ๐›ธ๐œ’ ๐›น๐œ“ ๐›บ๐œ”