expressed as a sum of ﬁve or less perfect squares

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expressed as a sum of ﬁve or less perfect squares

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Source: CroatianTST 2006 and 18-th KoreanMO 2005(Final Round)

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#1 h May 9, 2007, 5:39 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Find all natural numbers that can be expressed in a unique way as a sum of ﬁve or less perfect squares.

This post has been edited 2 times. Last edited by N.T.TUAN, May 12, 2007, 9:44 AM

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Rust

#2 h May 9, 2007, 6:10 PM • 2 Y

Y by Adventure10, Mango247

Only $0$ , because $1=0^{2}+1^{2}=1^{2}+0^{2}$ .

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#3 h May 10, 2007, 4:58 AM • 2 Y

Y by Adventure10, Mango247

But I see that $2$ also is a solution of problem.
Note:
1)Parts of sum must are positive integers.
2) $1^{2}+2^{2}$ and $2^{2}+1^{2}$ are same.
Now try again!

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ZetaX

#4 h May 10, 2007, 7:32 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Two well known and already posted theorems:
a) every integer can be written as sum of four squares.
b) every integer $n>N$ (with e.g. $N=169$ ) can be written as sum of five nonzero squares.

Thus every big integer can be written in two ways, so we are left to check some small numbers, done.

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#5 h May 10, 2007, 10:11 AM • 1 Y

Y by Adventure10

ZetaX wrote:

b) every integer $n>N$ (with e.g. $N=169$ ) can be written as sum of five nonzero squares.

Wow, please post that link here, thanks

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ZetaX

#6 h May 10, 2007, 10:39 AM • 2 Y

Y by Adventure10, Mango247

http://www.mathlinks.ro/Forum/viewtopic.php?t=121595

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#7 h May 12, 2007, 8:04 AM • 4 Y

Y by bobaboby1, Adventure10, Mango247, and 1 other user

Hello,

ZetaX wrote:

Two well known and already posted theorems:
a) every integer can be written as sum of four squares.
b) every integer $n>N$ (with e.g. $N=169$ ) can be written as sum of five nonzero squares.

Thus every big integer can be written in two ways, so we are left to check some small numbers, done.

I think that if it is possible to use four squares theorem, it's useless to use b) and check up to 169. There is something simpler :

For $n\geq 16$ , if n can be written as $a^{2}+b^{2}+c^{2}+d^{2}$ with either 1, either 2, either 3, either 4 not in the set $\{a,b,c,d\}$ , then n can be written as $u^{2}+v^{2}+w^{2}+t^{2}+x^{2}$ where $x$ is the missing number (1,2,3 or 4) and $(u^{2},v^{2},w^{2},t^{2})$ the 4 or less squares of expression of $n-x^{2}$ . Because $x$ is not in in the set $(a,b,c,d)$ , the two expressions are different.

So we have just to check numbers 1 to 15 plus the number $30=1^{2}+2^{2}+3^{2}+4^{2}$ .

$1=1$ is OK
$2=1+1$ is OK
$3=1+1+1$ is OK
$4=1+1+1+1=4$ is NOK
$5=4+1=1+1+1+1+1$ is NOK
$6=4+1+1$ is OK
$7=4+1+1+1$ is OK
$8=4+4=4+1+1+1+1$ is NOK
$9=9=4+4+1$ is NOK
$10=9+1=4+4+1+1$ is NOK
$11=9+1+1=4+4+1+1+1$ is NOK
$12=9+1+1+1=4+4+4$ is NOK
$13=9+1+1+1+1=4+4+4+1$ is NOK
$14=9+4+1=4+4+4+1+1$ is NOK
$15=9+4+1+1$ is OK

$30=16+9+4+1=25+4+1$ is NOK

And the requested set is $\{1,2,3,6,7,15\}$ (+ $0$ is $0$ is to be considered)

A question now is : can this problem be solved without using four squares theorem ?

--
Patrick

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