ppllpd_aux.nb

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$pvn[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2p, 2q}$

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${(-n^2 + p^2 + q^2)/n, 2 p, 2 q}$

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$Factor[len2[{(p^2 + q^2 + r^2 - n^2)/(n), 2p, 2q, 2r}]]$

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$Factor[len2[{((2p + 1)^2 + (2q + 1)^2 + (2r + 1)^2 - n^2)/(2n), 2p + 1, 2q + 1, 2r + 1}]]$

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(n^2 + 4 p^2 + 4 q^2 + 4 r^2 + 4 p + 4 q + 4 r + 3)^2/(4 n^2)

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( 0 1 1 1 ) 1 0 1 1 1 1 0 1 1 1 1 2

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$pvnx[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2p, 2q, (p^2 + q^2 + n^2)/n}$

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${(-n^2 + p^2 + q^2)/n, 2 p, 2 q, (n^2 + p^2 + q^2)/n}$

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${2 p + 2 q + (n^2 + p^2 + q^2)/n, 2 q + (-n^2 + p^2 + q^2)/n + (n^2 + p^2 + q^2)/n, 2 p + (-n^ ... + p^2 + q^2)/n + (n^2 + p^2 + q^2)/n, 2 p + 2 q + (-n^2 + p^2 + q^2)/n + (2 (n^2 + p^2 + q^2))/n}$

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${2 p + 2 q + (n^2 + p^2 + q^2)/n, 2 q + (-n^2 + p^2 + q^2)/n + (n^2 + p^2 + q^2)/n, 2 p + (-n^2 + p^2 + q^2)/n + (n^2 + p^2 + q^2)/n}$

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$pvnx2[p_, q_, n_] := {(p^2 + q^2 - n^2)/(n), 2p, 2q, -(p^2 + q^2 + n^2)/n}$

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( 0 1 1 1 ) 1 0 1 1 1 1 0 1 1 1 1 2

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( 0 1 1 1 ) -1 0 1 1 -1 1 0 1 -1 1 1 2

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( 0 -1 1 1 ) 1 0 1 1 1 -1 0 1 1 -1 1 2

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( 0 1 -1 1 ) 1 0 -1 1 1 1 0 1 1 1 -1 2

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( 0 -1 -1 1 ) 1 0 -1 1 1 -1 0 1 1 -1 -1 2

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( 0 -1 1 1 ) -1 0 1 1 -1 -1 0 1 -1 -1 1 2

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( 0 1 -1 1 ) -1 0 -1 1 -1 1 0 1 -1 1 -1 2

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( 0 -1 -1 1 ) -1 0 -1 1 -1 -1 0 1 -1 -1 -1 2

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( 0 1 1 -1 ) 1 0 1 -1 1 1 0 -1 -1 -1 -1 2

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( 1 0 0 0 ) 0 1 0 0 0 0 1 0 0 0 0 1

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( 1 0 0 0 ) 0 1 0 0 0 0 1 0 0 0 0 1

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( 1 0 0 0 ) 0 1 0 0 0 0 1 0 0 0 0 1

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( 1 0 0 0 ) 0 1 0 0 0 0 1 0 0 0 0 1

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( 0 0 0 0 ) 0 0 0 0 0 0 0 0 0 0 0 0

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Factor[len2[J123 . pvnx[p, q, n]] - len2[pvnx[p, q, n]]] 

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$Prop5u[p_, q_] := {4q^2 (p^2 + q^2 - 1), 8p q^2, 8q^3}$

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$Prop5v[p_, q_, r_] := {4 (r^2 - 1) q^2 + (2 p r + 1)^2, 8 q^2 r, -4q (2 p r + 1)}$

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$u := {25 p^2 - 25 q^2, 50 p q, 0} v := {25 p^2 - 25 q^2, -50 p q, 0} w := {0, -28 p q, 96 p q}$

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Created by Mathematica (August 11, 2005)