Prove the following identities:

Let and be any real numbers

Then

can be confirmed algebraically.

This identity has an interesting application to the Pell family.

Let be an integer sequence satisfying the Pell recurrence.

Suppose we let and

Then

This gives the abovementioned Pell identities.

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The RHS, 2 [ (Pn)^2 + 2(Pn)(Pn+1) + 4(Pn+1)^2 ]^2, when fully expanded, gives:

2(Pn)^4 + 8(Pn)^3(Pn+1) + 24(Pn)^2(Pn+1)^2 + 32(Pn)(Pn+1)^3 + 32(Pn+1)^4

Each term in the Pell Sequence is given by (Pn) = 2(Pn-1) + (Pn-2) , so therefore (Pn+1) = 2(Pn) + (Pn-1) , and (Pn+2) = 2(Pn+1) + (Pn) .

Substituting 2(Pn+1) + (Pn) for (Pn+2), the LHS, (Pn)^4 + 16(Pn+1)^4 + (Pn+2)^4 , can now be re-written, as

= (Pn)^4 + 16(Pn+1)^4 + (2Pn+1 + Pn)^4

= (Pn)^4 + 16(Pn+1)^4 + (Pn)^4 + 8(Pn)^3(Pn+1) + 24(Pn)^2(Pn+1)^2 + 32(Pn)(Pn+1)^3 + 16(Pn+1)^4

= 2(Pn)^4 + 8(Pn)^3(Pn+1) + 24(Pn)^2(Pn+1)^2 + 32(Pn)(Pn+1)^3 + 32(Pn+1)^4 = LHS

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