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Do there exist such three different positive integers $a, b, c$ that the following numbers: $a+b, b+c, c+a, a+b+c$ are all squares of integers and three of them ($a+b, b+c, c+a, a+b+c$) are squares of consecutive integers?

My first thought is to order the numbers $a,b,c$. Let $a<b<c$ and $a+b=k^2$, $b+c=l^2$, $c+a=m^2$, $a+b+c=n^2$. Then $n^2>l^2>m^2>k^2$ so $n>l>m>k$. Now we can consider two variants:

(i) $\quad n=l+1=k+2$
(ii) $\quad l=k+1=l+2$

I don't know what to do from this place...

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  • $\begingroup$ I'm sorry if it hurt You. I didn't mean to do it. I'm rewriting my try just now. I'll put it here in a minute. $\endgroup$ – Roy C. Apr 30 '17 at 13:49
  • $\begingroup$ My first thought is to order the numbers $a, b, c$. Let $a<b<c$ and $a+b=k^2, b+c=l^2, c+a=m^2, a+b+c=n^2$. Then $n^2>l^2>m^2>k^2$ so $n>l>m>k$. Now we can consider two variants: 1. $n=l+1=k+2$ 2. $l=k+1=l+2$ I don't know what to do from this place... $\endgroup$ – Roy C. Apr 30 '17 at 13:53
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    $\begingroup$ Thanks for adding that, Roy; it's really helpful (and respectful) to the members of the community. I've edited your work into the question. $\endgroup$ – Théophile Apr 30 '17 at 14:00
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    $\begingroup$ Thank you very much. As you see, I'm new and must get into the habits :) $\endgroup$ – Roy C. Apr 30 '17 at 14:06
  • $\begingroup$ artofproblemsolving.com/community/… artofproblemsolving.com/community/… $\endgroup$ – individ Apr 30 '17 at 14:14
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In the previous post, i miss understood the question.

We need to find $a,b,c$ such that $a+b,a+c,b+c,a+b+c$ all are square number and three of them are consecutive square numbers.

Which means with $d,e$ integers $a+b,a+c,b+c,a+b+c$ are equal to $(d-1)^2,d^2,(d+1)^2 ,e^2$. since $a,b,c$ are positive so $a+b+c >a+b,a+c,b+c$ which means that $a+b+c =(d+1)^2 $ or $a+b+c = e^2$, and since the other three equations are symmetric it does not matter how to order them, so with out loss of generality let :

$a+b=(d-1)^2$ and $a+c = d^2$ and $b+c=(d+1)^2$ and $a+b+c=e^2$.

Summing togther the first three equations we get $a+b+a+c+b+c = 2(a+b+c)= 2e^2=(d-1)^2+d^2 +(d+1)^2$.

Solving the equation $2e^2 = (d-1)^2 + d^2 +(d+1)^2$ we arrive at $2e^2=3d^2+2$

Solving this equation in Wolfram for integers gives that for any positive integer number $n$ the following formulas :

$d_n = \frac{\left(5+2 \sqrt{6}\right)^n-\left(5-2 \sqrt{6}\right)^n}{\sqrt{6}}$ and $e_n = \frac{1}{2} \left(\left(5-2 \sqrt{6}\right)^n+\left(5+2 \sqrt{6}\right)^n\right) $

Now substituting these values instead of $d,e$ in the equations $a+b=(d-1)^2,a+c=d^2,b+c=(d+1)^2,a+b+c=e^2$ gives that for any positive integer number $n$ the following formulas :

$a_n=\sqrt{\frac{2}{3}} \left(5-2 \sqrt{6}\right)^n+\frac{1}{12} \left(5-2 \sqrt{6}\right)^{2 n}-\sqrt{\frac{2}{3}} \left(5+2 \sqrt{6}\right)^n+\frac{1}{12} \left(5+2 \sqrt{6}\right)^{2 n}-\frac{1}{6}$

$b_n=\frac{1}{12} \left(5-2 \sqrt{6}\right)^{2 n}+\frac{1}{12} \left(5+2 \sqrt{6}\right)^{2 n}+\frac{5}{6}$

$c_n=-\sqrt{\frac{2}{3}} \left(5-2 \sqrt{6}\right)^n+\frac{1}{12} \left(5-2 \sqrt{6}\right)^{2 n}+\sqrt{\frac{2}{3}} \left(5+2 \sqrt{6}\right)^n+\frac{1}{12} \left(5+2 \sqrt{6}\right)^{2 n}-\frac{1}{6}$.

For the first few integers $n$ we get the following triples :

$n=0$ => $\{0,1,0\}$

$n=1$ => $\{0,9,16\}$

$n=2$ => $\{720,801,880\}$

$n=3$ => $\{77616, 78409, 79200\}$.

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