<s>[INST] Integer solutions to nonlinear system of equations $(x+1)^2+y^2 = (x+2)^2+z^2$ and $(x+2)^2+z^2 = (x+3)^2+w^2$ 
Do there exist integers $x,y,z,w$ that satisfy \begin{align*}(x+1)^2+y^2 &= (x+2)^2+z^2\\(x+2)^2+z^2 &= (x+3)^2+w^2?\end{align*}

I was thinking about trying to show by contradiction that no such integers exist. The first equation gives $y^2 = 2x+3+z^2$ while the second gives $z^2 = 2x+5+w^2$. How can we find a contradiction from here?
 [/INST] The differences between consecutive squares are consecutive odd numbers.  So we have:
$y^2-z^2=$ odd number
$z^2-w^2=$ next odd number
Of course if these were reversed (if the word "next" were moved one equation up) then the solution would be simple; any three consecutive numbers would work.  As it is, it's trickier, but certainly not impossible.
Phrasing the equations as I have, a little thought will reveal that any sequence of consecutive odd numbers which can be partitioned such that the smaller numbers add up to an odd number that is $2$ greater than the sum of the larger numbers of the partition, will provide a solution.
What's more, any solution to the original equation will exhibit the characteristic described in the preceding paragraph; the two questions are equivalent.

Borrowing from another existing answer, let's check this solution against my statement above:
$$31^2 + 12^2 = 32^2 + 9^2 = 33^2 + 4^2$$
The odd numbers which comprise the difference between $4^2$ and $9^2$ are $$(4+5), (5+6),(6+7),(7+8),(8+9)$$ or in other terms: $$9,11,13,15,17$$
The odd numbers which comprise the difference between $9^2$ and $12^2$ are $$19,21,23$$
The first five numbers of course add up to $13\times5=65$, and the latter three numbers add up to $21\times3=63$, which illustrates my earlier conclusion.

And, of course, $65$ and $63$ are the differences between $(x+2)^2$ and $(x+3)^2$, and $(x+1)^2$ and $(x+2)^2$, respectively, which tells us that $x$ will equal half of the smaller of these two odd numbers, after the extra $3$ units are subtracted:  $$\frac {63-3} 2=30$$  (This is because the smaller number, $63$, equals $(x+1)+(x+2)$.)
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