Silverman

I would like to share just one or two more results from my equation-hacking program before taking a short break.

The Silverman Constant is a prime-related number about which little is known. It derives from two functions: the Totient Function and the Divisor Function. The Totient of an integer is the number of prime numbers less than or equal to it, while the Divisor function is the sum of all the factors it has, including both 1 and itself. The Totient of 10, for example, is 3, because 1, 5 and 7 are the three prime numbers less than it, while the Divisor of 10 is 18, because its factors are 1, 2, 5, and 10, and 1+2+5+10=18.

Both of the above functions were discovered by Leonhard Euler in the 18th century, and have become useful in the theory behind both designing and breaking encryption algorithms.

Needless to say, the history of this constant has been difficult to track down. As best I have been able to gather, based upon an informal suggestion in a 1990s newsgroup, a mathematician in 2007 found that the infinite sum of the reciprocals of the products of the two functions converged upon a single number, which was named the Silverman Constant. Most of the literature surrounding its discovery is shrouded in mystery though, with many of the publications pertaining to it having been taken down.

The program I ran stumbled upon a close approximation of this number, of which only a few digits are known. The equation is:

Silverman Constant ~= (ln(Pi^e) / ln(2) - e^(Pi^2 / (12 * ln(2))))^3

{Pi = 3.14159265358979323846264338327950288; e = 2.71828182845904523536028747135266250}

As with previous findings, the above equation when translated to mathematical notation is:

Sm ~= (e * log2(π) - eπ2/(12 * ln(2)))3

Sm = 1.786576574653699218416745896828380038174941606202769599962

Again, I cannot stress enough that the above equation, as with the others, is based largely upon conjecture. I will provide no mathematical proof, due to the simple fact that it is beyond my capabilities. This could very well be a simple mathematical oddity, which we will learn if/when more decimal places of the constant are unearthed. If there is an accomplished mathematician out there who would like to fiddle with these numbers and try to make sense out of them, they are more than welcome to do so.

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