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The Engineer

Join Date: Feb 2005
Location: Albany, New York
Posts: 5060

# Expressing Positive Integers as the Sum of Squares

10/28/2006 7:05 PM

French mathematician Joseph-Louis Lagrange proved that every positive integer is either a square itself or the sum of two, three, or four squares. No more than four squares, x2 + y2 + z2 + t2, are ever needed to express any number, no matter how large.

Given Lagrange's result, number theorists asked whether there are other such expressions, called quadratic forms, that also represent all positive integers. In 1916, Indian mathematician Srinivasa Ramanujan uncovered 53 such expressions. He showed, for instance, that every number could be written as a square plus twice a square plus three times a square plus five times a square.

Now a Mathematician named Manjul Bhargava from Princeton has shown with a simplified calculation, that altogether there are 204 universal, matrix-defined quadratic forms. Mathematicians previously thought that the tally had been completed in 1948, when Margaret F. Willerding in her doctoral research at St. Louis University painstakingly worked out 178 universal matrix-defined quadratic forms. Bhargava's new enumeration, taking advantage of the shortcut provided by the 15 theorem, shows that Willerding missed 36 universal forms, listed one universal form twice, and included nine forms that are, in fact, not universal.

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Anonymous Poster
#1

### Re: Expressing Positive Integers as the Sum of Squares

10/30/2006 1:10 AM

X^2+Y^2+Z^2+T^2 ?

But what about 1111 its need 5 number to be squared.

33^2 + 4^2 + 2^2 + 1^2 + 1^2 =1111

Can anybody explain where I mistaken?

Anonymous Poster
#3

### Re: Expressing Positive Integers as the Sum of Squares

10/30/2006 3:18 AM

here is the solution for 1111 = sum if sq(31,10,7,1) but I would repeat hasan's question. Can anybody tell where we can apply this?

for any number or interval of numbers you can simply find the solution by an easy programme by GUI programming langugues.

Member

Join Date: Jul 2006
Posts: 6
#2

### Re: Expressing Positive Integers as the Sum of Squares

10/30/2006 1:14 AM

very interesting... but I was wondering where we could use this.. as in a practical use of it... or how is it superior to counting from 0 to "positive infinity"..? sorry if I have over looked something or if this is just too dumb to ask..