<s>[INST] Can this problem have solutions for $N>3$:Find three $(N=3)$ positive integers a, b and c such that: $a+b+c=k^2$, $a+b=t^2$, $b+c=m^2$ and $a+c=n^2$ Can this question have solutions for N>3:
Find three ($N =3$) positive integers a, b and c such that: $a+b+c=k^2$, $a+b=x^2$, $b+c=m^2$ and $a+c=n^2$; $x, m, n∈\mathbb   N$
Solution:
Let $a+b+c=(x+1)^2=x^2+2x+1$
If $a+b=x^2$ then third number $c=2x+1$.
If $b+c=(x-1)^2=x^2-2x+1=m^2$ then :
$a=4x$, so $b=x^2-4x$
Now due to statement we must have $a+c=6x+1=n^2$
$n^2=6x+1$ can have infinite solutions such as:
$(x, n^2)=(20, 121=11^2), (60, 361=19^2), (140, 841=29^2)\cdot\cdot\cdot$
Which give:
$(a, b, c, k)=(80, 320, 41, 21), (240, 3360, 121, 61), (560, 19040, 281, 141),\cdot\cdot\cdot$
I tried to solve this problem for $N=4$, four numbers but no success. Now I have two questions:
1-Does this problem have solutions for N>3?
2-Any idea for better method?
 [/INST] The solution given by "@ player3236" has $k=325$.
The integer $325$ can be written as sum of two squares
in three way's & is shown below:
$k= x_1^2+x_2^2=x_3^2+x_4^2=x_5^2+x_6^2$
($x_1,x_2,x_3,x_4,x_5,x_6$)=$(15,10,17,6,18,1)$
$325= 15^2+10^2=17^2+6^2=18^2+1^2$
Above can help in arriving at a parametric solution.
Also mathematician "Seiji Tomita has given parametric
solution in which $(a,b,c,d)$ taken two at a time is a
square & the link is given below.
http://www.maroon.dti.ne.jp/fermat/dioph115e.html
And click on (dioph115e)
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