Wait, doesn't that prove that any number one less than a square is composite? And four less, and nine less, etc.?
Yes, it does seem that a number that appears a square amount before a square will always be composite, given that the square root of that distance isn't one less than the square root of the square in question.
Even if a = n - 1, it seems 2n - 1 could still very well be composite. Take the square 400, for example: n = 20 in that case, so n - 1 would be 19. 2n - 1 would be 39, and that is obviously divisible by 3.
In a way, it seems that squares are like anti-prime mines: a prime can't be found trailing certain distances behind a square...