Gauss guessed
that Nature used a dice to decide if the number
N is prime. But how many sides are there on the
dice. Look again at the table showing the proportion
of primes amongst all numbers:
| N |
Number of primes from 1 up to
N, often referred to as 
(N) |
On average, how many numbers
you need to count before you expect a prime
number |
| 10 |
4 |
2.5 |
| 100 |
25 |
4.0 |
| 1,000 |
168 |
6.0 |
| 10,000 |
1,229 |
8.1 |
| 100,000 |
9,592 |
10.4 |
| 1,000,000 |
78,498 |
12.7 = 10.4+2.3 |
| 10,000,000 |
664,579 |
15.0 = 12.7+2.3 |
| 100,000,000 |
5,761,455 |
17.4 = 15.0+2.3 |
| 1,000,000,000 |
50,847,534 |
19.7 = 17.4+2.3 |
| 10,000,000,000 |
455,052,511 |
22.0 = 19.7+2.3 |
Every time Gauss multiplied N by
10, the number of sides on the prime number dice
(shown in the last column) to test primes around
N goes up by adding approximately 2.3.
A function is like a computer programme: you
feed a number into the function and it calculates
away and spits out an answer.
Here Gauss feeds the prime number dice function
with a number N and the function outputs how many
sides are on the dice to choose primes around
N. Every time you multiply the input by 10, you
add 2.3 to the output.
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