2.4 Case n = 5.When n = 5, ( 2.3) reduces to
The roots of ( 2.43) are of the form
Expanding ( 2.44) gives
Consider the co-efficient of x3 in ( 2.45). In ( 2.46) after multiplying out this reduces to
Comparing ( 2.47) with the coefficient of x3 in ( 2.43) gives
Consider the co-efficient of x2 in ( 2.45). In ( 2.46) after multiplying out this reduces to
Comparing this to the co-efficient of x2 in ( 2.43) yields
Substituting ( 2.49) for the coefficient of p in ( 2.52) gives
Eq.( 2.54) may be rewritten as
Now substitution from ( 2.49) gives
and dividing through by 20a(a - b) gives
The numerator in the quotient of ( 2.57) is of the same order in a and b as the
denominator and is therefore a pure number, q5. Thus
Now, substituting ( 2.59) into the positive root of ( 2.44) yields
and clearly, in this case for x to be positive, q5 must be less than
unity. Re-arranging ( 2.60) for a
It is clear from ( 2.61) that 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 q5 exhibiting satisfactory characteristics, this proves Fermat's Last Theorem for n = 5.
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P.G.Bass, April 2009
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