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 \(x^2y^2-x^2-y^2+2\) where the positive integers \(x\) and \(y\) are not consecutive,
  3. It cannot be expressed in the form \(x^2y^2-x^2-y^2+2\) for any consecutive positive integers \(x\) and \(y\).

The first partridge number is between \(1\) and \(2014\). What number is it?

This problem is part of the set Advent Calendar 2014.


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