I would have found the explanation much clearer if you stated at the outset whether the length of the line is chosen by us or for us. As is, the explanation has a lot of parts that are allowed to move from the outset: the location where we are standing, the length of the line extending behind us, and the number of primes on that line. Since the goal is to impose a relationship between these three components, and ultimately our standing location is the only bit that is allowed to vary, it would have helped me make things more concrete if you started with something like “Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This sets up the three key questions for the theorem: how far does the line extend backwards, what number does it whisper, and what does that number have to do with primes?
“Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This is a beautiful sentence. I wish I’d thought of it :D I’ll see if I can find my own version, because it strikes a lovely balance between leaving the specific details for later in the article, while building anticipation for them.
I would have found the explanation much clearer if you stated at the outset whether the length of the line is chosen by us or for us. As is, the explanation has a lot of parts that are allowed to move from the outset: the location where we are standing, the length of the line extending behind us, and the number of primes on that line. Since the goal is to impose a relationship between these three components, and ultimately our standing location is the only bit that is allowed to vary, it would have helped me make things more concrete if you started with something like “Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This sets up the three key questions for the theorem: how far does the line extend backwards, what number does it whisper, and what does that number have to do with primes?
This is a beautiful sentence. I wish I’d thought of it :D I’ll see if I can find my own version, because it strikes a lovely balance between leaving the specific details for later in the article, while building anticipation for them.
Feel free to use it wholesale.