Day 6: A Fun Game

Probability Level 5

A mathematician writes the integers 1,2,3,4,,20151,2,3,4,\ldots,2015 on a whiteboard, with which he plays a game.

Every turn he selects two numbers aa and bb that are on the whiteboard, subject to two conditions:

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

Then he rubs off aa and bb and replaces them with a2b+2a4b2a^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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