On 22 September 2017 AD, the feast of St Maurice.

Andrew Yanthar-Wasilik

Finding General Formula for Equations from the previous book, “Book 5 Integer Formula for Dimensionless Coupling Constants of Fundamental Forces”.

All Coupling Constants use the same equations, from which constants can be calculated.

✠Part A of the equation (A/B)C

Integer Formula for fine structure constant alpha, αE, ruling electromagnetic force is

A16 = ( C0)(24 ⁄ 24)

Integer Formula for the weak force, αW, ruling force of decays is

A17 = ( C0)(27 ⁄ 24)

Integer Formula for the strong nuclear force, αS, ruling quarks, and nucleons is

A1 = ( C0)( − 21 ⁄ 24)

From this partial sequence can be obtained a complete sequence.

At n=8, there is an axis of symmetry. Axis at n=8 divide (as it will be shown on the Graphs of the Function) values between Real and Complex. This can be achieved only when the denominator equals 24.

For n=8, the nominator equals 0.

For an explanation of how to find formulae for some sequences, see the webpage of Brian Kell(1)

So we have in the first row number of a constant, in the second-row value of the exponent of “A,”

and in the third row, the difference between the two adjoining values of the exponent “A”.

n =               11                                      12                                13                                 14                            15                                16                                 17

E(n) =       (9)/(24)               (12)/(24)              (15)/(24)            (18)/(24)          (21)/(24)             (24)/(24)             (27)/(24)

Δ(E(n)) =  (3)/(24)                (3)/(24)               (3)/(24)               (3)/(24)              (3)/(24)                     (3)/(24)

Constant difference between the values of the Exponents Δ(E(n)) = (3)/(24) and this is the slope ( (y)/(Δx))= a ) of the function (y = a*x + b) we are trying to find. Y-intercept equals to b. Y-intercept is at C0 and equals b = -(24)/(24).

So, the exponent can be written as a linear function:

y = ((3)/(24))*x + (-(24)/(24)) = (3*x − 24)/(24) = (x − 8)/(8)

Now, the “A” part of the equation can be written as:

✠Ax = ( C0)((x − 8)/(8))     (Eqn. A)

✠Part B

The next sequence is the “B” part of the equation (A/B)C

B16 = ( C16 / ( 8 + 2 * (24/24) ) )(88 ⁄ 24)

B17 = ( C17 / ( 9 + 2 * ( 24/27 ) ) )(99 ⁄ 24)

B1 = ( C1 / ( -7 + 2 * ( -24/21 ) ) )( − 77 ⁄ 24)

Taking under consideration the Exponent part first - it can be seen that this sequence is the one in “A” multiplied by a factor of 11, giving:

n =                          6                               7                           8                                 9                        10                        11

E (n ) =           -(22)/(24)          -(11)/(24)        (0)/(24)            (11)/(24)        (22)/(24)          (33)/(24)

Δ(E(n)) =         (11)/(24)             (11)/(24)       (11)/(24)           (11)/(24)      (11)/(24)

Slope, a = y)/(Δx) = (11)/(24)

Y-intercept, at C0, equals b = -(88)/(24)

And formula for Exponent EB sequence is:

EB= ((11*x − 88)/(24))

Sequence of the base of the power is:

Constant C16 is ( 8 + 2 * (24/24)

Constant C17 is ( 9 + 2 * ( 24/27 )

Constant C1 is ( -7 + 2 * ( -24/21 )

It can be seen that in the 2 * d part, the “d” equals to reciprocal of the value of the exponent from part A, which we calculated right at the beginning so that this sequence will be equal to:

EpartB = 2 * ((8)/(x − 8)) = ((16)/(x − 8))

Remaining factor from the sum:

8, 9, -7 gives sequence = (x-8)

So, the base of the power equals to:

Base of the power = (x-8) + ((16)/(x − 8)) = (x2 − 16x + 80)/(x − 8)

If we take the reciprocal of this function, we will have the product of Cand this reciprocal raised to the Exponent EB. Constants in the nominator of the base of the power C16, C17, C1, etc., are just the number of the constant, “n”, which we replaced by “x”, and each constant can be calculated from the formula from “Book 1”.

Cx = ( C0) ( π/ e )x     (Eqn. I)

Formula for sequence of the part “B” is then:

✠Bx = {[(C0)(πe)x][(x − 8)/(x2 − 16x + 80)]}[(11x − 88)/(24)]     (Eqn. B)

✠Part C of the equation (A/B)C

The Exponents of the equation (A/B)C are:

The exponent of the constant C16 is ( C16 ) * (8/24)

The exponent of the constant C17 is ( C17 ) * (9/24)

The exponent of the constant C1 is ( C1 ) * (-7/24)

The sequence : -(7)/(24),..., (8)/(24), (9)/(24),... has an equation equal to ((1)/(3)) of the first one in this article, i.e.,

yc = ((1)/(3)) * ( (x − 8)/(8) ) = (x − 8)/(24)

And CEx = ( C0) ( π/ e )x is the same as in the Eqn. I

(It should be Cx instead of CEx . But that would be mixed up with the final result.)

Product of ( CEx) and ( yc = (x − 8)/(24) ) is the Exponent Cxwe are looking for:

✠Cx = ( ( C0) ( π/ e )x ) * ((x − 8)/(24))      (Eqn. C)

And the final result of Exponent Main, ExpM is ExpM = ( A / B )C

Substituting the Equations A, B, C, and we get the long formula for ExpM, which is way too long to write; therefore, I will use parts A, B, C of the Equation

ExpM = ( A / B )C      (Eqn. EM)

In part II, I will derive the General Formula for any value of the constant alpha and similar to alpha.

Andrew Yanthar-Wasilik