8
$\begingroup$

Let $c_n \in \{ -1,1\}$. Here, it is stated that every natural number may be written as

$$\sum c_kk^2$$

Where $k$ runs from $1$ to some finite number. I am wondering whether every natural number $n$ can be written as follows:

$$n = \sum c_n(2k-1)^2.$$

In other words,

can every natural number be written as the sum of the first so-and-so signed odd squares?

Obviously, $1=1^2$. However, even to find such a writing of $2$, I needed eight squares: $$2=1+9+25-49+81-121-169+225$$ And could not find one for $3$. Any insight would be appreciated.

$\endgroup$
2
  • 6
    $\begingroup$ For any four consecutive odd squares $a^2-(a+2)^2-(a+4)^2+(a+6)^2=16$, so if we can make 1 to 16 inclusive then we're done. $\endgroup$ – nickgard Mar 17 '20 at 2:39
  • 4
    $\begingroup$ If you can show that any $n \in [0, 8]$ can be written in that form, then any $n \in [-7, -1]$ can also be written (reverse the signs), and therefore any $n \in [-7, 8]$ can be expressed as desired. Then induction would follow: If $n$ can be expressed, then $$n + 16 = n +(2k+7)^2 - (2k+5)^2-(2k+3)^2 + (2k+1)^2$$ can be expressed as well. $\endgroup$ – Catalin Zara Mar 17 '20 at 2:39
7
$\begingroup$

How high do you have to go? $$1=+1^2$$ $$2=+1^2+3^2+5^2-7^2+9^2-11^2-13^2+15^2$$ $$3=+1^2+3^2+5^2+7^2-9^2$$ $$4=-1^2-3^2-5^2-7^2+9^2-11^2-13^2+15^2-17^2+19^2$$ $$5=+1^2+3^2+5^2+7^2-9^2+11^2+13^2+15^2+17^2-19^2-21^2$$ $$6=-1^2-3^2+5^2-7^2-9^2+11^2$$ $$7=+1^2+3^2+5^2+7^2+9^2+11^2+13^2+15^2+17^2-19^2+21^2-23^2-25^2-27^2+29^2$$ $$8=-1^2+3^2$$ $$9=-1^2-3^2+5^2-7^2-9^2-11^2-13^2-15^2-17^2+19^2-21^2-23^2-25^2-27^2+29^2+31^2+33^2$$ $$10=+1^2+3^2$$ $$11=-1^2-3^2+5^2-7^2-9^2-11^2-13^2-15^2+17^2-19^2-21^2+23^2+25^2$$ $$12=-1^2-3^2-5^2-7^2-9^2+11^2-13^2+15^2$$ $$13=-1^2-3^2-5^2-7^2+9^2+11^2-13^2-15^2+17^2$$ $$14=-1^2-3^2-5^2+7^2$$ $$15=-1^2-3^2+5^2$$ $$16=+1^2-3^2-5^2+7^2$$ $$17=+1^2-3^2+5^2$$ $$18=+1^2+3^2-5^2+7^2-9^2+11^2+13^2-15^2$$ $$19=+1^2+3^2+5^2-7^2+9^2+11^2-13^2$$ $$20=-1^2-3^2+5^2-7^2-9^2-11^2-13^2-15^2+17^2+19^2$$ $$21=+1^2+3^2+5^2+7^2+9^2+11^2+13^2-15^2-17^2-19^2+21^2$$ $$22=+1^2-3^2+5^2-7^2-9^2-11^2-13^2-15^2+17^2+19^2$$ $$23=+1^2+3^2+5^2+7^2+9^2+11^2+13^2-15^2+17^2+19^2-21^2+23^2+25^2-27^2-29^2$$ $$24=-1^2+3^2+5^2-7^2-9^2+11^2$$ $$25=-1^2-3^2-5^2-7^2-9^2-11^2+13^2+15^2-17^2-19^2-21^2-23^2-25^2-27^2+29^2+31^2+33^2$$ $$26=+1^2+3^2+5^2-7^2-9^2+11^2$$ $$27=-1^2-3^2-5^2-7^2-9^2-11^2+13^2-15^2+17^2-19^2+21^2$$ $$28=+1^2+3^2+5^2+7^2+9^2-11^2-13^2+15^2+17^2-19^2$$ $$29=+1^2-3^2-5^2-7^2-9^2-11^2+13^2-15^2+17^2-19^2+21^2$$ $$30=-1^2+3^2-5^2-7^2-9^2+11^2-13^2+15^2$$

$\endgroup$
2
  • 3
    $\begingroup$ No higher. See the comments to the OP by myself and Catalin Zara. $\endgroup$ – nickgard Mar 17 '20 at 2:45
  • $\begingroup$ @nickgard Cool. This program runs pretty much instantaneously. Do you guys have more fun challenges while it's in my view? $\endgroup$ – user5713492 Mar 17 '20 at 2:49
6
$\begingroup$

This is possible if $c_k \in \{-1, 0, 1\}$. For each $m$, consider that $$1 = (2m + 1)^2 + (m^2 + m - 1)^2 - (m^2 + m + 1)^2,$$ which is a sum/difference of odd squares. Note also that the minimal square, $(2m + 1)^2$, can be made as large as we want, hence we can make $1$ as many times as we want without reusing squares. Thus, to make any $n$, we simply need to make $1$ $n$ times, out of different odd squares, and sum the result.

(As a bonus, we can make each $n$ out of odd squares in this way infinitely many ways, as we can subtract $1$s as well as we like.)

$\endgroup$
2
  • 1
    $\begingroup$ Would you be using all squares from 1 to $\max$? Note that $c_k \in \{ -1, 1\}$. $\endgroup$ – Catalin Zara Mar 17 '20 at 2:31
  • $\begingroup$ I failed to notice that condition. I'll leave my answer here, given the partial results already posted. $\endgroup$ – user759562 Mar 17 '20 at 2:32
2
$\begingroup$

Shortest representations of small positive integers.

\begin{align*} 1 &= 1 \\ 2 &= 1 + 9 + 25 - 49 + 81 - 121 - 169 + 225 \\ 3 &= 1 + 9 + 25 + 49 - 81 \\ 4 &= -1-9-25-49+81-121-169+225-289+361 \\ 5 &= 1+9+25+49-81+121+169+225+289-361-441 \\ 6 &= -1-9+25-49-81+121 \\ 7 &= 1+9+25-49+81+121+169+225+289+361+441+529-625-729-841 \\ &= 1+9+25+49+81+121+169+225+289-361+441-529-625-729+841 \\ 8 &= -1+9 \\ 9 &= -1-9-25-49-81+121-169-225-289-361-441-529+625+729+841+961-1089 \\ &= -1-9+25-49-81-121-169-225-289+361-441-529-625-729+841+961+1089 \\ 10 &= 1 + 9 \\ 11 &= -1-9+25-49-81-121-169-225+289-361-441+529+625 \\ 12 &= -1-9-25-49-81+121-169+225 \\ 13 &= -1-9-25-49+81+121-169-225+289 \\ 14 &= -1-9-25+49 \\ 15 &= -1-9+25 \\ 16 &= 1-9-25+49 \\ 17 &= 1-9+25 \\ 18 &= 1+9-25+49-81+121+169-225 \\ 19 &= 1+9+25-49+81+121-169 \\ 20 &= -1-9+25-49-81-121-169-225+289+361 \\ 21 &= 1+9+25+49+81+121+169-225-289-361+441 \\ 22 &= -1-9+25-49-81+121+169-225-289+361 \\ &= -1+9-25-49+81-121-169+225-289+361 \\ &= 1-9+25-49-81-121-169-225+289+361 \\ 23 &= 1+9+25+49+81+121+169-225+289+361-441+529+625-729-841 \\ 24 &= -1-9+25+49+81-121 \\ &= -1+9+25-49-81+121 \\ 25 &= -1-9-25-49-81-121-169-225-289-361+441+529+625+729-841+961-1089 \\ &= -1-9-25-49-81-121-169+225-289-361-441+529-625+729+841+961-1089 \\ &= -1-9-25-49-81-121-169-225+289-361+441-529+625-729+841-961+1089 \\ &= 1-9-25-49-81-121-169-225-289-361-441-529+625+729+841-961+1089 \\ &= -1-9-25-49-81-121+169-225-289-361+441-529+625-729-841+961+1089 \\ &= -1-9-25-49+81-121-169-225-289-361-441+529+625-729-841+961+1089 \\ &= -1-9-25-49-81-121+169+225-289-361-441-529-625-729+841+961+1089 \\ 26 &= 1-9+25+49+81-121 \\ &= 1+9+25-49-81+121 \\ 27 &= -1-9-25-49-81-121+169-225+289-361+441 \\ 28 &= -1+9+25+49+81+121+169+225-289-361 \\ &= 1+9+25+49+81-121-169+225+289-361 \\ 29 &= -1-9-25-49+81-121+169-225+289+361-441 \\ &= -1-9+25-49-81-121-169+225+289+361-441 \\ &= -1-9-25+49-81+121-169-225+289-361-441 \\ &= 1-9-25-49-81-121+169-225+289-361+441 \\ 30 &= -1-9-25+49+81-121-169+225 \\ &= -1+9-25-49-81+121-169+225 \\ 31 &= -1-9+25+49-81-121+169 \\ 32 &= -1+9-25+49 \\ 33 &= -1+9+25 \\ 34 &= 1+9-25+49 \\ 35 &= 1+9+25 \\ 36 &= -1-9-25-49-81-121+169+225+289-361 \\ 37 &= -1+9+25+49+81+121+169=225-289+361+441-529-625 \\ &= 1+9+25+49+81-121-169+225+289+361+441-529-625 \\ &= 1+9+25+49+81+121-169+225-289-361+441+529-625 \\ 38 &= -1-9-25+49-81+121-169+225+289-361 \\ &= 1-9-25-49-81-121+169+225+289-361 \\ &= -1+9+25-49-81-121-169-225+289+361 \\ 39 &= 1+9+25+49+81+121+169+225-289+361+441-529-625 \\ 40 &= -1-9+25-49+81+121+169-225+289-361 \\ &= 1-9-25+49-81+121-169+225+289-361 \\ &= -1-9+25+49+81-121+169-225-289+361 \\ &= -1+9+25-49-81+121+169-225-289+361 \\ &= 1+9+25-49-81-121-169-225+289+361 \end{align*}

$\endgroup$
1
  • $\begingroup$ Oops, you are right, I did indeed misunderstand the constraints. $\endgroup$ – orlp Mar 17 '20 at 18:33
1
$\begingroup$

$$\,\,\,\,\,\,\,\,\,\,\,$$ $$3=1+9+25+49-81$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.