$P$ terms are perfect squares and $S$ arithmetic progression of 3 perfect squares.
$S_1=P_1,P_2,P_3$ such that $P_3-P_2=P_2-P_1$
$S_2=P_4,P_5,P_6$ such that $P_6-P_5=P_5-P_4$
$S_3=P_7,P_8,P_9$ such that $P_9-P_8=P_8-P_7$
and $P_7-P_6=P_4-P_3$

Apologies for any incorrect usage of terms, any edit is welcome.

  • Trivial cases: All equal $0$ or $1$. – farruhota Jun 24 '17 at 5:34
  • Can there be solutions with distinct numbers? – agentrsdg Jun 24 '17 at 5:37
  • 1
    $$(49, 169, 289) (5041, 7225, 9409) (14161, 34225, 54289) \\ (49, 169, 289) (1681, 7225, 12769) (14161, 34225, 54289) \\ (49, 289, 529) (2209, 4225, 6241) (7921, 22201, 36481)$$ – Paul LeVan Jun 24 '17 at 7:18
  • 1
    $1^2,5^2,7^2; 7^2,13^2,17^2; 17^2,85^2,119^2.$ – farruhota Jun 24 '17 at 7:47

Better system to write this.

$$\left\{\begin{aligned}&{P_3}^2-{P_2}^2={P_2}^2-{P_1}^2\\&{P_6}^2-{P_5}^2={P_5}^2-{P_4}^2\\&{P_9}^2-{P_8}^2={P_8}^2-{P_7}^2\\& {P_7}^2-{P_6}^2={P_4}^2-{P_3}^2 \end{aligned}\right.$$

For simplicity, the formula is written. Solution write using other parameters which will be determined next.

$$P_1=X^2-2XY-Y^2$$

$$P_2=X^2+Y^2$$

$$P_3=(P^2+2PS-S^2)K^2-2(P^2-2PS-S^2)KT-(P^2+2PS-S^2)T^2=$$

$$=X^2+2XY-Y^2$$

$$P_4=(P^2+S^2)(K^2-2KT-T^2)$$

$$P_5=(P^2+S^2)(K^2+T^2)$$

$$P_6=(P^2+S^2)(K^2+2KT-T^2)$$

$$P_7=(P^2-2PS-S^2)K^2+2(P^2+2PS-S^2)KT-(P^2-2PS-S^2)T^2=$$

$$=A^2-2AB-B^2$$

$$P_8=A^2+B^2$$

$$P_9=A^2+2AB-B^2$$

Express these options through the following polynomial.

$$X=Py^2-2Sxy+(P+2S)x^2+2(P^2-2PS-S^2)tx-P(P^2+2PS-S^2)t^2$$

$$Y=(2P-S)y^2+2Pxy-Sx^2+2(P^2-2PS-S^2)ty-S(P^2+2PS-S^2)t^2$$

$$K=y^2-2xy-x^2-(P^2+2PS-S^2)t^2$$

$$T=2((P+S)x+(P-S)y+(P^2-2PS-S^2)t)t$$

$$A=(2S-P)a^2+2Sab-Pb^2+2(P^2+2PS-S^2)at+P(P^2-2PS-S^2)t^2$$

$$B=Sa^2-2Pab+(2P+S)b^2+2(P^2+2PS-S^2)bt+S(P^2-2PS-S^2)t^2$$

Express these options through the following polynomial.

$$y=e^2+2(2q-e)Fn+z^2-2zq-q^2$$

$$x=e^2+2(2z-e)Fn-3z^2+2qz-q^2$$

$$a=e^2+2e(Fn-2z)+z^2+2qz-q^2$$

$$b=e^2-2(e+2z)Fn+z^2+2qz-q^2+8(P^2-S^2)n^2$$

$$t=4n(Fn-z)$$

And now Express them via the parameters set by us $ - p,s,k,j,c$.

$$F=4ps$$

$$P=s^2+2p^2$$

$$S=s^2-2p^2$$

$$z=F(k^2-j^2)+2(P-S)Fcj-2(P-S)((P+S)F+4PS)c^2$$

$$n=k^2-j^2-2(P^2-S^2)c^2$$

$$q=8PScj-2Fj^2$$

$$e=F(k^2-2kj-j^2)+8PSkc+2(P-S)Fcj-2(P-S)((P+S)F+4PS)c^2$$

The formula is very cumbersome and I wasn't able to test it. We must believe not made any mistake?

More simple parameterizations.

$$P_1=(q^2-2ql-l^2)(p^2-2ps-s^2)((k^2-2kt-t^2)x^2+2(k^2+2kt-t^2)xy-$$

$$-(k^2-2kt-t^2)y^2)$$

$$P_2=(q^2+l^2)(p^2-2ps-s^2)((k^2-2kt-t^2)x^2+2(k^2+2kt-t^2)xy-$$

$$-(k^2-2kt-t^2)y^2)$$

$$P_3=(q^2+2ql-l^2)(p^2-2ps-s^2)((k^2-2kt-t^2)x^2+2(k^2+2kt-t^2)xy-$$

$$-(k^2-2kt-t^2)y^2)$$

$$P_4=(k^2-2kt-t^2)(q^2+2ql-l^2)(p^2-2ps-s^2)(x^2+y^2)$$

$$P_5=(k^2+t^2)(q^2+2ql-l^2)(p^2-2ps-s^2)(x^2+y^2)$$

$$P_6=(k^2+2kt-t^2)(q^2+2ql-l^2)(p^2-2ps-s^2)(x^2+y^2)$$

$$P_7=(p^2-2ps-s^2)(q^2+2ql-l^2)((k^2+2kt-t^2)y^2+2(k^2-2kt-t^2)xy-$$

$$-(k^2+2kt-t^2)x^2)$$

$$P_8=(p^2+s^2)(q^2+2ql-l^2)((k^2+2kt-t^2)y^2+2(k^2-2kt-t^2)xy-$$

$$-(k^2+2kt-t^2)x^2)$$

$$P_9=(p^2+2ps-s^2)(q^2+2ql-l^2)((k^2+2kt-t^2)y^2+2(k^2-2kt-t^2)xy-$$

$$-(k^2+2kt-t^2)x^2)$$

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