Appendix A

Derivation of All Solutions for n = 2.

When n = 2, ( 2.1) exhibits an infinite number of solutions. All of these can, theoretically, be derived as follows.

When n = 2, ( 2.3) reduces to

x² - 2( a - b )x - ( a² - b² ) = 0

(A.1)

Solving for x using the standard formula,

x = ( a - b ) ą { 2a( a - b ) }^{1 / 2}

(A.2)

For simplicity, as before, put p = (a- b), so that (A.2) becomes

x = p ą ( 2ap )^1/2

(A.3)

Thus with a and b, and therefore p, as integers, for x to be an integer, only even squares can appear under the root sign in (A.3). Accordingly, there is a solution to Fermat's equation for n = 2 whenever 2pa is an even square. When 2pa > 4, there are multiple solutions for each even square. The following table lists all the solutions for the first six even squares.

#	2pa	p	a	b	x	y	z	x²	y²	z²
1	4	1	2	1	3	4	5	9	16	25
2	16	1	8	7	5	12	13	25	144	169
3	16	2	4	2	6	8	10	36	64	100
4	36	1	18	17	7	24	25	49	576	625
5		2	9	7	8	15	17	64	225	289
6		3	6	3	9	12	15	81	144	225
7	64	1	32	31	9	40	41	81	1600	1681
8		2	16	14	10	24	26	100	576	676
9		4	8	4	12	16	20	144	256	400
10	100	1	50	49	11	60	61	121	3600	3721
11		2	25	23	12	35	37	144	1225	1369
12		5	10	5	15	20	25	225	400	625
13	144	1	72	71	13	84	85	169	7056	7225
14		2	36	34	14	48	50	196	2304	2500
15		3	24	21	15	36	39	225	1296	1521
16		4	18	14	16	30	34	256	900	1156
17		6	12	6	18	24	30	324	576	900
18		8	9	1	20	21	29	400	441	841

Table A.1 - Pythagorean Triples for the First Six Even Squares of 2pa.

As shown by the caption to Table A.1, these solutions are, for obvious reasons, known as Pythagorean Triples. It is also obvious that not all such triples are unique. In the above Table for instance, entries 3, 6, 8, 9, 12, 14, 15, 16 and 17 are multiples of lower triples. Those that are unique are known as Principle Triples, and are distinguished by having prime numbers for x and/or z.

The second point to note in the Table is that, both entries 4 and 12 can also be represented in the form

x² + y² = z⁴

(A.4)

which raises the question as to how many other Pythagorean triples can be so represented. It also raises the question as to whether Fermat's equation can be further generalised to

x^l + y^m = zⁿ

(A.5)

and whether integer solutions exist for this equation, or any of its possible other variants.

Of interest is a version of (A.5) thus

x² + y² = z³

(A.6)

which, with z = x + a, converts to the variable co-efficient elliptic curve

y² = x³ + ( 3a - 1 )x² + 3a²x + a³

(A.7)

for which integer solutions would, in the light of Wiles' proof of the Taniyama - Shimura Conjecture incur associated Modular Forms.

In the method used here, (A.6) also converts to the variable co-efficient 3^rd order polynomial in x

x³ + ( 3a - 2 )x² + ( 3a² - 2b )x + (a³ - b²) = 0

(A.8)

The search for integer solutions would not be as simple because identification of the form of the roots is not straightforward.

REFERENCES.

[1] Harold M. Edwards, Fermat's Last Theorem - A Generic Introduction to Algebraic Number Theory. Springer - Verlag, 1977.

[2] Simon Singh, Fermat's Last Theorem, Harper Perennial, 1997.

M2 Version 1.0.0
Ó P.G.Bass, April 2009

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