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Splines, Points and Fermat

Abstract. Spline plots and embedded integer values of cn-an-bn are used to graphically depict examples where Fermat is correct.

Fermat's general equation is

an+bn=cn    (1)

where 0<a<b<c, n=2,3,4,...

and

a=2,3,4,...,    (2)

b=a+1,2,3,...,    (3)

c=b+1,2,3,...    (4)

We define "sum" to be

sum=cn-an-bn.    (5)

We can plot the points for sum vs. a "perturbation index". Fig.1 shows such a plot. (The Power Basic program that computes the points is provided in Appendix A.)

Figure 1 Sum Points Plotted

Clarity is enhanced if we join the plots with a spline curve. Fig 2 shows the spline/sum plot.

Figure 2 Spline Curve with Embedded Sums

Table 1 is a list of values plotted in Fig. 2.

Table 1

 Index Sum a b c 1 3 4 9 10 2 24 4 9 11 3 5 4 10 11 4 28 4 10 12 5 7 4 11 12 6 32 4 11 13 7 -4 5 10 11 8 19 5 10 12 9 -2 5 11 12 10 23 5 11 13 11 0 5 12 13 12 27 5 12 14 13 -13 6 11 12 14 12 6 11 13 15 -11 6 12 13 16 16 6 12 14 17 -9 6 13 14 18 20 6 13 15

Index, Sum, a, b and c in Fig. 2.

As Fig. 2 shows, the curve crosses sum=0 at 11 points on the plot. However, according to Table 1 only one point corresponds to the all-integer solution a=5, b=12, c=13. And we see that this point is distinguished from the others by two attributes: (1) The spline curve is tangent to the sum=0 ordinate axis; (2) The tangent point is not a point of inflection. Paraphrasing Fermat, only points on the curve that satisfy these two requirements correspond to all-integer solutions of Eq. 1.

What about values of n>2? Fig. 3 shows the plot for typical values of sum in

a3+b3=c3. (6)

Figure 3 n=3

Note in this case how the curve appears to coincide with or cross the sum=0 axis at 4 points. And 2 of these points look like they could possibly meet the tangent/no point of inflection requirement. However, an examination of Table 2 reveals that the two points in question are not quite tangents to the sum=0 axis. Thus no point on the plot constitutes an all-integer solution of Fermat's equation.

Table 2

 Index Sum a b c 1 -398 9 10 11 2 -1 9 10 12 3 -332 9 11 12 4 137 9 11 13 5 -603 10 11 12 6 -134 10 11 13 7 -531 10 12 13 8 16 10 12 14 9 -862 11 12 13 10 -315 11 12 14 11 -784 11 13 14 12 -153 11 13 15

Index, Sum, a, b and c in Fig. 3.

According to Fermat's theorem, there are in general no curves (for n>2) that meet the tangent/no point of inflection requirement. How Fermat knew that this is so, for an infinitude of combinations of a, b and c and for all n>2, remains a mystery! He reportedly wrote that he had a proof which was a little too lengthy to write in the margin of a book he was reading. The world has never found or rediscovered such a little proof in the centuries since his death. Andrew Wiles produced a proof, but it is 200 pages long and uses ideas not conceived of in Fermat's day.

The search continues ...

Appendix A

#COMPILE EXE
#DIM ALL

FUNCTION PBMAIN () AS LONG
DIM n AS LONG
n=2
DIM iteration AS LONG
DIM a AS LONG 'Lowest a=2
DIM b AS LONG 'b=a+1+,,,
DIM c AS LONG 'c=b+1+...
DIM sum AS LONG 'c^n-b^n-a^n
OPEN "c:\users\Marjorie Dixon\Documents\plotsx1.dat" FOR OUTPUT AS #3 'a values
OPEN "c:\users\Marjorie Dixon\Documents\plotsy1.dat" FOR OUTPUT AS #4 'b values
OPEN "c:\users\Marjorie Dixon\Documents\plotsz1.dat" FOR OUTPUT AS #5 'c values
OPEN "c:\users\Marjorie Dixon\Documents\plotssum.dat" FOR OUTPUT AS #2 'sum values
OPEN "c:\users\Marjorie Dixon\Documents\plotsiteration.dat" FOR OUTPUT AS #1 'index
iteration=0
FOR a=4 TO 6
FOR b=a+5 TO a+7
FOR c=b+1 TO b+2
sum=c^n-b^n-a^n
iteration=iteration+1
WRITE #1,iteration
WRITE #2,sum
WRITE #3,a
WRITE #4,b
WRITE #5,c
NEXT c
NEXT b
NEXT a
CLOSE #1 'iteration
CLOSE #2 'sum
CLOSE #3 'a
CLOSE #4 'b
CLOSE #5 'c
MSGBOX "End of program"