2.5 Case n =7.When n =7, ( 2.3) reduces to
The Roots of ( 2.62) are of the form
Expanding ( 2.63) gives
In ( 2.64) consider the co-efficient of x5. From ( 2.65) and ( 2.66), after multiplying out this reduces to
Now consider the co-efficient of x4 in ( 2.64). From ( 2.65) and ( 2.66),
after multiplying out this reduces to
Comparing ( 2.68) with the co-efficient of x4 in ( 2.62) gives
In ( 2.70) substitute for the co-efficient of (a - b) from ( 2.67) and reduce to
Dividing the term in the k's by (k1+k2+k3) yields
Now substitution from ( 2.67) yields
and dividing through by 42a(a-b) then gives
In ( 2.75), the numerator in the quotient is of the same order in a and b as the denominator and therefore the quotient will be a pure number, q7.
Therefore, ( 2.75) becomes
Substitution of ( 2.77) into the positive root of ( 2.63) then gives
and as in the previous case, q7 must be less than unity for x to be
positive. Re-arranging for a
and thus a cannot be an integer following the same argument as in Section 2.2. Therefore z cannot be an integer in ( 2.1). Thus, subject to q7 exhibiting satisfactory characteristics, this proves Fermat's Last Theorem for n = 7. 2.6 Extrapolation to n = Any Odd Number.From the results for n = 5 and n = 7, although only two cases are involved, the analytical process being completely rigorous, permits extrapolation from these two cases to the general one of n = any odd number. Thus for the general case
characteristics, this, in conjunction with the result for n = 4, proves Fermat's Last Theorem for all n.
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P.G.Bass, April 2009
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