All 3 experiments refers to topics explained in the paper.
Experiment #1 : Will be done with the context of Euler sums, we will usea vector containing 8 elements (8 real numbers with 64 digits precision). Those 8 numbers are suspected to be related to Sa(2,3), an unknown number X, which is believed to be related by a integer relation identity. See Euler sums in the present paper.
We did not know exactly how Sa(2,3) could be related to those 8 elements,
we had a series and at least it was possible to get the actual floating
point value of X, that is : 0.15616693338117...
The 8 elements were : Zeta values and polylogarithms of high order
evaluated at 1/2 :
Sa(2,3), Li 5(1/2), Li 4(1/2)log(2), log(2)^5, Zeta(5), Zeta(4)log(2),
Zeta(3)log(2)^2, Zeta(2)log(2)^3, Zeta(2)Zeta(3)
Experiment #2 refers to the section about
New Formulas for Pi...
In this experiment we have a set of values that are all coming from the same type of series, simple sums of inverses. This time the unknown constant is very well known , what we don't know is IF this particular type of series can represent the number Pi.
Experiment #3 refers to the section about
Applications of the PSLQ algorithm
We will test here if a particular real number is algebraic or not. Of
course the answer can't be given completely since we test only small
degrees with relatively small coefficients. This kind of experiment tells
us if that particular number X is a root of a polynomial with integer
coefficients, in other words, if X is algebraic.
We will test if the number is a root of a polynomial of degree 5 or less
with small coefficients. The real experiment used PSLQ up to degree 25 with
much larger coefficients.
In this case the experiment returned an answer which is interpreted as being
not good..., it failed to find a short answer.
The thing we can say from this is that : Up to coefficients of size
approximately 9000, there seems to be no polynomial that has
this X has a root.