Day 6: A Fun Game

A mathematician writes the integers \(1,2,3,4,\ldots,2015\) on a whiteboard, with which he plays a game.

Every turn he selects two numbers \(a\) and \(b\) that are on the whiteboard, subject to two conditions:

  • If one of \(a\) and \(b\) are prime, he will always let \(a\) be prime; so if he picks 5 and 8, then he will let \(a = 5, b= 8\).
  • But if both are prime, or both are not prime, then \(a \leq b\)

Then he rubs off \(a\) and \(b\) and replaces them with \(a^2b+2a-4b-2\).

He continues this process until one number is left on the whiteboard.

What number is this?

Note: By primes I do only mean positive primes only.

This problem is part of the Advent Calendar 2015.

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