Sums of Triangular Numbers

[davar55]

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.)

[R. Gerbicz]

Originally Posted by davar55

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).

[alpertron]

By the way you can find the decomposition in sum of three squares using my applet on: http://www.alpertron.com.ar/FSQUARES.HTM

[mfgoode]

Originally Posted by davar55

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

[axn]

Originally Posted by mfgoode

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.

[R.D. Silverman]

Originally Posted by mfgoode

: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.

[xilman]

Originally Posted by R.D. Silverman

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

[mfgoode]

Originally Posted by xilman

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

[wblipp]

Originally Posted by mfgoode

whch really matters, is am I right or wrong?

I guess you mean:

Originally Posted by mfgoode

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

[mfgoode]

Originally Posted by wblipp

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]

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