## New answers tagged arithmetic-progression

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 ...

0

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 $(p_0;s_0)$ ...

Top 50 recent answers are included