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Can it be shown that

$$\sum_{q=0}^{u}(n+qd)^{2}\ne 3^t \ \ \ \ \forall n,d,u,t\in\mathbb{N}$$

Where we let $\mathbb{N}$ denote positive integers.

I am not confident there is no counterexample.

Edit, My attempted

If $u=4k-1$ then $\sum_{q=0}^{u}(n+qd)^{2}\ne 3^t $

Formula

$$ \sum_{q=0}^{u}(n+qd)^{2} =n^2(u+1)+d(2n+d)\frac{(u+1)u}{2} +d^2\frac{(u+1)u(u-1)}{3} \ \ \ \ \ \ ...eq(1)$$

Proof

Let's suppose

$\sum_{q=0}^{u}(n+qd)^{2}=3^t $

By $eq(1)$ we can write

$2×3^{t+1}=6n^2(u+1)+3d(2n+d)(u+1)u+2d^2(u+1)u(u-1)$

Now consider $u=4k-1$

$\implies 3^{t+1}=12n^2k+6d(2n+d)k(4k-1)+8d^2k(4k-1)(2k-1) $

We know

$12n^2k=even$

$6d(2n+d)k(4k-1)=even$

$8d^2k(4k-1)(2k-1)=even$

And $even +even +even =even\ne 3^{t+1}$

It's show complete proof for $u=4k-1$

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  • 1
    $\begingroup$ You body question is more than just a "sum of squares"; it a sum of progressive squares. Not sure if the result is different but it gives you more info. $\endgroup$
    – fleablood
    Commented Oct 10, 2019 at 18:00
  • 3
    $\begingroup$ quite, otherwise we could have things like $1^2+4^2+8^2=3^4$, but this example doesn't have the squares "evenly spaced." For a sum of just two squares, it seems the only possibility is $0^2+(3^n)^2 = 3^{2n}$. $\endgroup$
    – JMoravitz
    Commented Oct 10, 2019 at 18:04
  • $\begingroup$ @OP. I changed the language a little bit. I think its more clear now but feel free to roll it back. $\endgroup$
    – Mason
    Commented Oct 10, 2019 at 18:11
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    $\begingroup$ @Mason $$ \sum_{q=0}^{u}(n+qd)^2=n^2(u+1)+d(2n+d)\frac{(u+1)u}{2} +d^2\frac{(u+1)u(u-1)}{3} $$ $\endgroup$
    – Pruthviraj
    Commented Oct 10, 2019 at 18:31
  • 1
    $\begingroup$ @Mason may be this link helpful math.stackexchange.com/q/3369713/647719 $\endgroup$
    – Pruthviraj
    Commented Oct 10, 2019 at 18:39

1 Answer 1

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+50
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There is no $(n,d,u,t)$ such that $$\sum_{q=0}^{u}(n+qd)^{2}=3^t\tag1$$

Proof :

Suppose that there is $(n,d,u,t)$ satisfying $(1)$ which is equivalent to $$(u+1)\left(6n^2+6ndu+u(2u+1)d^2\right)=2\cdot 3^{t+1}$$

Let us separate it into three cases. Note here that $u+1\gt 1$ and $6n^2+6ndu+u(2u+1)d^2\gt 2$.

  • Case 1 : $(u+1,6n^2+6ndu+u(2u+1)d^2)=(2,3^{t+1})$. Then,$$n^2+(n+d)^2=3^t\tag2$$We see that both $n$ and $n+d$ have to be divisible by $3$, so setting $n=3n_1$ and $n+d=3d_1$ gives$$n_1^2+d_1^2=3^{t-2}\tag3$$Comparing $(3)$ with $(2)$, we see that if $t$ is even, then there are positive integers $N,D$ such that $N^2+D^2=1$ which is impossible. If $t$ is odd, then there are positive integers $N,D$ such that $N^2+D^2=3$ which is impossible.

  • Case 2 : $(u+1,6n^2+6ndu+u(2u+1)d^2)=(2\cdot 3^a,3^b)$ where $a,b$ are positive integers such that $a+b=t+1$. Then,$$6n^2+6nd(2\cdot 3^a-1)+(2\cdot 3^a-1)(4\cdot 3^a-1)d^2=3^b\tag4$$Setting $d=3d_1$ gives$$2n^2+2n\cdot 3d_1(2\cdot 3^a-1)+(2\cdot 3^a-1)(4\cdot 3^a-1)\cdot 3d_1^2=3^{b-1}$$Setting $n=3n_1$ gives$$6n_1^2+6n_1d_1(2\cdot 3^a-1)+(2\cdot 3^a-1)(4\cdot 3^a-1)d_1^2=3^{b-2}\tag5$$Comparing $(5)$ with $(4)$, we see that if $b$ is odd, then there are positive integers $a,N,D$ such that$$6N^2+6ND(2\cdot 3^a-1)+(2\cdot 3^a-1)(4\cdot 3^a-1)D^2=3$$which is impossible since the LHS is larger than $3$. If $b$ is even, then there are positive integers $a,N,D$ such that $$6N^2+6ND(2\cdot 3^a-1)+(2\cdot 3^a-1)(4\cdot 3^a-1)D^2=1$$which is impossible since the LHS is larger than $1$.

  • Case 3 : $(u+1,6n^2+6ndu+u(2u+1)d^2)=(3^a,2\cdot 3^b)$ where $a,b$ are positive integers such that $a+b=t+1$. Then, $$6n^2+6nd(3^a-1)+(3^a-1)(2\cdot 3^a-1)d^2=2\cdot 3^b\tag6$$Setting $d=3d_1$ gives$$2n^2+2n\cdot 3d_1(3^a-1)+(3^a-1)(2\cdot 3^a-1)\cdot 3d_1^2=2\cdot 3^{b-1}$$Setting $n=3n_1$ gives$$6n_1^2+6n_1d_1(3^a-1)+(3^a-1)(2\cdot 3^a-1)d_1^2=2\cdot 3^{b-2}\tag7$$Comparing $(7)$ with $(6)$, we see that if $b$ is odd, then there are positive integers $a,N,D$ such that$$6N^2+6ND(3^a-1)+(3^a-1)(2\cdot 3^a-1)D^2=6$$which is impossible since the LHS is larger than $6$. If $b$ is even, then there are positive integers $a,N,D$ such that $$6N^2+6ND(3^a-1)+(3^a-1)(2\cdot 3^a-1)D^2=2$$which is impossible since the LHS is larger than $2$.

From the three cases above, the conclusion written at the top follows.

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  • $\begingroup$ Thank you for your answe. please explain, In case$1$, how to compare $(3)$ with $(2)$ and conclude $N^2+D^2=1,3$? $\endgroup$
    – Pruthviraj
    Commented Oct 16, 2019 at 16:08
  • $\begingroup$ @Pruthviraj: Sure. The key is the fact that both $(2)$ and $(3)$ are of the form $a^2+b^2=3^s$. Take $t=10$ for example. Suppose that we have $n^2+(n+d)^2=3^{10}$. Then, we have to have $n=3n_1$ and $n+d=3d_1$, so $n_1^2+d_1^2=3^8$. Again, we have to have $n_1=3n_2$ and $d_1=3d_2$, so $n_2^2+d_2^2=3^6$. Again, we have to have $n_2=3n_3$ and $d_2=3d_3$, and so on... eventually we have to have $N^2+D^2=1$ which is impossible. If $t$ is odd, then we have $N^2+D^2=3$ which is also impossible. This idea is also used in Case 2 and Case 3. $\endgroup$
    – mathlove
    Commented Oct 16, 2019 at 17:20
  • $\begingroup$ Okay... one more doubt, how, if $n^2+(n+d)^2=3^t$ then $3|n$ and $3|(n+d)$. is necessary? $\endgroup$
    – Pruthviraj
    Commented Oct 16, 2019 at 19:19
  • $\begingroup$ @Pruthviraj: If you check all possible cases, then you'll know that it is necessary that $3\mid n$ and $3\mid (n+d)$. If $3\mid n$ and $3\not\mid (n+d)$, then $n^2+(n+d)^2$ is not divisible by $3$. If $3\not\mid n$ and $3\mid (n+d)$, then $n^2+(n+d)^2$ is not divisible by $3$. If $3\not\mid n$ and $3\not\mid (n+d)$, then $n^2+(n+d)^2$ is not divisible by $3$. If $3\mid n$ and $3\mid (n+d)$, then $n^2+(n+d)^2$ is divisible by $3$. So, it is necessary that $3\mid n$ and $3\mid (n+d)$. $\endgroup$
    – mathlove
    Commented Oct 17, 2019 at 4:07
  • $\begingroup$ What do you think about generalizing on this problem, any suggestions (ideas)? $\endgroup$
    – Pruthviraj
    Commented Oct 17, 2019 at 17:41

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