2.7 The q Numbers.
To consolidate the proof, it is necessary to show that the q numbers exhibit
satisfactory characteristics as x, b and n independently explore their theoretical
maxima and minima, and therefore do not cause anomalies in the relationships
developed for the a's.
2.7.1 The q Numbers in Cases n = 3 and 4.
These cases are unique in that as indicated in Sections 2.2 and 2.3, these q
numbers must always be greater than unity.
From ( 2.19) or ( 2.39), re-arranging for q
(a) Minimum Values of x and b.
The minimum values of x and b are each unity. Substitution into ( 2.81) then
gives
As b < a < 2, q must be greater than unity. Note that, because all three
independent variables have been specified here, it is possible to calculate a
from ( 2.2) for the two cases involved. Thus for n = 3, a = 1.08008 and for n = 4, a
= 1.03054.
(b) Maximum x, (b must be unity).
The maximum value of x is x ® ¥. Dividing ( 2.81) throughout by x
gives
and as x ® ¥, q ® 1 from a value > 1.
(c) Maximum b, (x = any value <
¥).
The maximum value of b is b ® ¥ which incurs a ® ¥ and
therefore as b ® ¥ ( 2.81) becomes
so that as b ® ¥, q ® ¥. Thus in the case of n = 3
and n = 4, the range of variation of q numbers do not cause an anomaly in the
relationships developed for the parameter a.
2.7.2 The q Numbers for n > 4.
In this case the q numbers must always be less than unity as shown in Sections
2.4 and 2.5. From ( 2.80), re-arranging for qn
|
qn = |
x - ( n - 1 ) a + nb/2
x - a + b
|
|
| (2.85) |
(a) Minimum Values.
The minimum values of x and b are each unity. Substitution into ( 2.85) then
gives
|
qn = |
1 + n/2 - ( n - 1 ) a
2 - a
|
|
| (2.86) |
The minimum value of n in this case is 5, which when substituted into ( 2.86)
gives
Because all three independent variables have been specified here, it is
possible to determine the value of a from ( 2.2). The value is 1.0123. Clearly,
when substituted into ( 2.87), qn < +1.
Also it is obvious from ( 2.2) that as n is increased, a reduces and as n ®
¥, a ® 1 so that in ( 2.86), qn ® -¥. Thus
in this case, qn does not therefore incur anomalies in the relationship
developed for a for all n > 4.
(b) Maximum x, (b must be unity).
The maximum value of x is x ® ¥. Dividing ( 2.85) throughout by x
gives
|
qn = |
1 - ( n - 1 )a/x + n /2x
|
|
| (2.88) |
so that as x ® ¥, qn ® 1 from a value < 1.
In this case as n ® ¥, n/x® 1 so that ( 2.88) becomes
and because a ® 1, qn < 1. Thus in this case qn does not incur
anomalies in the relationship developed for a for all n > 4.
(c) Maximum b, (x = any value < ¥).
The maximum value of b is b ® ¥. Dividing ( 2.85) throughout by b
gives
qn = |
|
x/b - |
( n - 1 ) a
b
|
+ n/2 |
1 + x/b - a/b
|
|
| (2.90) |
As b ® ¥, so a ® ¥ and ( 2.90) then becomes
Clearly this result is unaffected by changes in n and so qn does not cause
anomalies in this case in the relationships developed for a for all n > 4.
(d) Maximum n, (x and b = any
value < ¥).
The maximum value of n is n ® ¥. Dividing ( 2.85) throughout by n
gives
|
qn = |
x/n - ( 1 - 1/n )a + b/2
( x - a + b ) /n
|
|
| (2.92) |
As n ® ¥, this becomes
Thus in this case also qn does not cause anomalies in the relationships
developed for a for all n > 4.
The overall result from this review of the q numbers, is that they incur no
anomalies in the relationships developed for the a's as the three independent
variables, x , b and n, independently explore their maxima and minima. Therefore,
the proof of Fermat's Last Theorem, presented herein, is sound for all
values of x, y and n.
M2 Version 1.0.0
Ó
P.G.Bass, April 2009
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