Magic Sine: The Prime Numbers (2006) by Roger Saravia Aramayo (All Rights Reserved)
During a lot of time I have been thinking and calculating about the prime numbers and looking for the way to create simple formulas in order to reveal them. One night I discovered that these numbers were related with the sine function. So starting from that, I produced several interesting expressions about the famous prime numbers.
Starting from the sine function we can construct a powerful base expression which allows the construction of interesting expressions related to the prime numbers. With two natural numbers a and b, this base expression would let us verify if b is factor of a: 0 if b is a factor of a sign sin 2 a / b 1 if b is not a factor of a
(1)
A prime number x is a natural x that doesn't have any factor between 2 and x 1 , inclusive. Applying the product of the expression (1) in this range, we have: 0 sign sin x / i 1 x 1
2
i 2
if x is not a prime number if x is a prime number
(2)
Now, we will modify starting from the expression (1) to invert its output and to let further on the construction of a more advanced expression than (2). Then: 0 if b is not a factor of a 1 sign sin 2 a / b 1 if b is a factor of a
(3)
We already remembered that a prime number x is a natural that doesn't have any factor between 2 and x 1 , inclusive. Applying the sum of the expression (3) in this range, we have: 0 1 sign sin x / i f x 1
2
i 2
if x is a prime number if x is not a prime and f is the number of factors of x
(4)
The graph (i1) corresponds to the expression (4) for the first thirty-five natural numbers. A couple of notes regarding this graph:
The points on the x axis are the prime numbers.
Any other point corresponds to a not prime x. For example, x 12 is associated with f 4 ; this indicates that 12 have four factors without the unit and 12 itself. And certainly, their four factors are: 2, 3, 4 and 6. 1