Day 1: A Partridge in a Number Tree

A partridge sits in its tree. However, this is not a pear tree; it is a number tree! It rummages around until it finds a special type of number- let us call them partridge numbers.

Define a partridge number to be a number such that:

  1. It is a square number,
  2. It can be expressed in the form x2y2x2y2+2x^2y^2-x^2-y^2+2 where the positive integers xx and yy are not consecutive,
  3. It cannot be expressed in the form x2y2x2y2+2x^2y^2-x^2-y^2+2 for any consecutive positive integers xx and yy.

The first partridge number is between 11 and 20142014. What number is it?

This problem is part of the set Advent Calendar 2014.

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