Prime obsession by Derbyshire J.
By Derbyshire J.
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Extra resources for Prime obsession
By the same rough logic, we can estimate the size of the N-th prime. Consider a range of numbers from 1 to K, for some big number K. If the count of primes in that range is C, then on average we should expect to find the first of those numbers at K ÷ C, the second at 2K ÷ C, the third at 3K ÷ C, and so on. The N-th will be around NK ÷ C, and the C-th, which is to say the last in this range, will be around CK ÷ C, which means, of course, K. Now, if the PNT is true, then the count C is actually K ⁄ log K, so that the N-th prime is actually around NK ÷ (K ⁄ log K), which is to say, around N log K.
I chose the arguments to go up by adding 1 each time and shall continue to do that, just for convenience. In this particular function, this makes the values multiply by 5. Of course, there is nothing special about 5. 5. Each gives me an exponential function. ” When you have a situation like this, in which a certain phenomenon (in this case an exponential function) can show up in many different ways, there is generally one way mathematicians prefer to represent the whole phenomenon. So it is here.
Here, for the time being at any rate, is my definition of a square root. The square root of N is the single non-negative number (if any) which, when multiplied by itself, gives N. VIII. Fortunately the exponential function doesn’t give any of these problems. You can cheerfully invert it to give you a function that, when you pick arguments going up by multiplication, gives you values going up by addition. Of course, as with exponential functions, there is a whole family of inverse functions, depending on the multiplier; and as with the exponential function, mathematicians much, much prefer the one that goes up in additions of 1 when the arguments go up in multiples of e.