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3

Michael Stoll has given a nice answer, but here is a 17th century argument. Let $a$, $b$, $c$, $d$ be an arithmetic progression with common difference $\Delta \ne 0$. Suppose that $$1 + a b = z^2_1, \quad 1 + a c = z^2_2, \qquad 1 + a d = z^2_3,$$ $$1 + b c = z^2_4, \quad 1 + b d = z^2_5, \qquad 1 + c d = z^2_6.$$ Consider the following ...


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I think it is better to solve a more formal task. We write the system. $$\left\{\begin{aligned}&ab+T=x^2\\&ac+T=y^2\\&bc+T=z^2\end{aligned}\right.$$ We need to find solutions $a,b,c$ - that was an arithmetic progression. This will help the solution of the equation Pell. $$p^2-3s^2=T$$ Knowing any solution of the equation Pell $(p_0;s_0)$ ...



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