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 ( 2.38), re-arranging for q


q = x + b/2
x - a + b
(2.81)

(a) Minimum Values of x and b.

The minimum values of x and b are each unity. Substitution into ( 2.81) then gives


q = 1 1/2
2 - a
(2.82)

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


q = 1 + 1/2x
1 -   ( a - 1 )
x
(2.83)

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


q = x + b/2
x
(2.84)

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


qn = 3 1/2 - 4a
2 - a
(2.87)

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
1 -   ( a - 1 )
x
(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


qn =  3/2 - a
1
(2.89)

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


qn ®   - n/2 + 1
0
 ® - ¥
(2.91)

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


qn ®  - a + b/2
0
 ® - ¥
(2.93)

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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