__Splines, Points and Fermat__

__Abstract.__ Spline plots and embedded
integer values of c^{n}-a^{n}-b^{n }are used to
graphically depict examples where Fermat is correct.

Fermat's general equation is

a^{n}+b^{n}=c^{n}
(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=c^{n}-a^{n}-b^{n}.
(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

a^{3}+b^{3}=c^{3}.
(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"