Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

for any positive integer $k$,there exsit $m\in N$ and $a_{i}\in N,i=1,2,\cdots,k$,such

(1): $a_{i}\neq a_{j},i\forall i\neq j$,

(2): $$a^2_{1}+a^2_{2}+a^2_{3}+\cdots+a^2_{k}=m^3$$

My idea: if $k=1$,then we let $a_{1}=1$,then $$a^2_{1}=1^3=m^3$$ if $k=2$, then we let $a_{1}=5,a_{2}=10$,then we have $$a^2_{1}+a^2_{2}=5^2+10^2=125=5^3=m^3$$ if $k=3$, note $$3^2+4^2+10^2=125=5^3$$ then let $a_{1}=3,a_{2}=4,a_{3}=10,m=5$

But follow I can't find it and How prove this problem ,Thank you

share|improve this question

1 Answer 1

Let $1^2+2^2+\cdots+k^2=m$ and multiply through by $m^2.$

share|improve this answer
oh!,It's nice !!+1 –  china math Mar 1 at 4:32
One can use any set of $k$ distinct squares on the left side, then define their sum as $m$, and "multiply through by $m^2$" as above to get more examples. Question left: what are all the solutions? –  coffeemath Mar 1 at 4:54

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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