Every integer can be expressed as a sum of three triangular numbers.
Given two integers N=a+b+c and M=x+y+z , how to express the product N*M as a sum of three triangular numbers. Example:
11=0+1+10 and 13=0+3+10 ; the product 11*13=143=10+28+105
How to write the product N*M=F(a,b,c,x,y,z) +G(a,b,c,x,y,z) +H(a,b,c,x,y,z) as three triangular numbers?
Dear Mohamed,
Fascinating question. Perhaps a better notation would be x = F(x) + S(x) + T(x), where F(x) > S(x) > T(x). You are then looking to write F(N*M), S(M*N) and T(M*N) explicitly in terms of M and N. It's not my area but I suspect that the triangular number statement is existential rather than algorithmic and that there is no formula for F, S and T.
Dear Peter,
Thank you for your answer. I agree with you the triangular number statement is essential but the algorithmic approach too . The computational thinking is every where. We need both theorital and algorithmic approach.
I am agree with Peter Samuels suggestion!
Good luck, and all the best !
Mirjana
Dear Peter,
Dear Mirjana,
Thank you for your suggestion but could you illustrate your suggestion with an example.
x= F(x) + G(x) + T(x) and F(N*M), G(N*M) , T(N*M)
Sincerely
Azzedine
Dear Mohamed:
What Peter has said is not that the proof is "essential", but that it is "existential". This means that, in the proof, the existence of a solution is proved, but an specific solution is never shown: the proof DOES NOT use an algorithm to compute the solution. So we only know that F(x), G(x), T(x) do exist, but not how to compute them in an easy way from the knowledge of x. So probably there is no direct formula to solve your problem either. You should make a program which implements a search, maybe using some tricks to speed it up.
Dear Jose,
Thank you for your answer.
I have already done the program it works but we look for a direct formula or symbolic expression.
Sincerely
Azzedine
Well, another thing you must take into account it that for "big" numbers the decomposition is not unique, but there are multiple ways of writing the number as a sum of three triangular ones. For example, 10=10+0+0 = 6+3+1. If there were a formula, maybe it would depend on which decompositions you choose among all the possibilities
Dear Jose,
Thank you very much. You are right but usually the second decomposition is choosen
For the suggestion of Peter we can write x= x(x+1)/2 - x(x-1)/2 +0.
Then we get F(x), S(x) and T(x).
F(M*N)=M*N(M*N+1)/2 and S(M*N)=-M*N(M*N-1)/2 and T(M*N)=0
Sincerely
Azzedine
Dear Jose,
Thanks a lot. You are right.
We replace by x*x= x(x+1)/2 + x(x-1)/2 +0
Regards
Azzedine
Dear Steve,
The sum of triangular numbers to 2 is
2=1+1+0
Regards
Azzedine
So the triangular factors of the result of triangular numbers 3 and 6 multiplied together would be 18 + 0 + 0?
Hello Mohamed, I think we could not write every integer as sum of three triangular numbers. For example 20 = ? + ? + ?.
Dear Adel, thanks to 0, each number is a sum of 3 triangular numbers (in 1 or more different manners). Here it is 10+10+0 = 20.
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