By reading the sentence of Fermat explaining that his margin was too small to give the explanation of its guess I based the hypothése that Fermat physically wanted to increase the size of his sheet.

I thus took a square of paper and proceeded to the cutting (on this left)

by reading its papers and his correspondence, it's seems that he's in habit of the mistery but he always founds cleverly solutions,

Simple even cut (multiple by 4)

Cutting a square in four strips and assemble thoses one as showned at the left side.

Origin square =(n*4)² named B² (fb²(n)) n=1 ,

(n x 4)² = 16 (= 4)

C=5, A = 3

B²=C²-A² ou A²=C²-B² => f(a) => f(c)-f(b)

for N = 4 (formulae gived for sample )

Calculation of B (ou A) :

Calculation de C : (new big square)

C = fc(n)

(4(1/4+1) , 4*(1+1/4), 5

Calculation of A (ou B)

A=fa(n)

N divided by4

Calculation of B (ou A) :

Calculation of C : (new big square)

C = fc(n)

Calculation de A (ou B)

A=fa(n)

N=6 ou impair * npair

Sample n=6:

C=9+1 = 10 or

Matrice:

N	C	B(or A)	A(or B)	C²	B²	A²	B²+A²
4	5	4	3	25	16	9	25
6	10	6	8	100	36	64	100
8	17	8	15	289	64	225	289
10	26	10	24	676	100	576	676
12	37	12	35	1369	144	1225	1369
14	50	14	48	2500	196	2304	2500
16	65	16	63	4225	256	3969	4225
18	82	18	80	6724	324	6400	6724
20	101	20	99	10201	400	9801	10201
22	122	22	120	14884	484	14400	14884
24	145	24	143	21025	576	20449	21025
26	170	26	168	28900	676	28224	28900
28	197	28	195	38809	784	38025	38809
30	226	30	224	51076	900	50176	51076
32	257	32	255	66049	1024	65025	66049
34	290	34	288	84100	1156	82944	84100

Coefficiented Division (k) from an even square divided by 4

n=1, k=2 = (4nκ)² , arranged side by side

Calculation of B

Calculation of C

Calculation of A

factorised by k

	K=1			K=2			K=3			K=4
N	C	B	A	C	B	A	C	B	A	C	B	A
4	5	3	4	10	6	8	15	9	12	20	12	16
6	10	8	6	20	16	12	30	24	18	40	32	24
8	17	15	8	34	30	16	51	45	24	68	60	32
10	26	24	10	52	48	20	78	72	30	104	96	40
12	37	35	12	74	70	24	111	105	36	148	140	48
14	50	48	14	100	96	28	150	144	42	200	192	56
16	65	63	16	130	126	32	195	189	48	260	252	64
18	82	80	18	164	160	36	246	240	54	328	320	72
20	101	99	20	202	198	40	303	297	60	404	396	80
22	122	120	22	244	240	44	366	360	66	488	480	88
24	145	143	24	290	286	48	435	429	72	580	572	96
26	170	168	26	340	336	52	510	504	78	680	672	104
28	197	195	28	394	390	56	591	585	84	788	780	112
30	226	224	30	452	448	60	678	672	90	904	896	120
32	257	255	32	514	510	64	771	765	96	1028	1020	128
34	290	288	34	580	576	68	870	864	102	1160	1152	136
36	325	323	36	650	646	72	975	969	108	1300	1292	144
38	362	360	38	724	720	76	1086	1080	114	1448	1440	152
40	401	399	40	802	798	80	1203	1197	120	1604	1596	160
42	442	440	42	884	880	84	1326	1320	126	1768	1760	168
44	485	483	44	970	966	88	1455	1449	132	1940	1932	176
46	530	528	46	1060	1056	92	1590	1584	138	2120	2112	184
48	577	575	48	1154	1150	96	1731	1725	144	2308	2300	192
50	626	624	50	1252	1248	100	1878	1872	150	2504	2496	200
52	677	675	52	1354	1350	104	2031	2025	156	2708	2700	208
54	730	728	54	1460	1456	108	2190	2184	162	2920	2912	216
56	785	783	56	1570	1566	112	2355	2349	168	3140	3132	224
58	842	840	58	1684	1680	116	2526	2520	174	3368	3360	232
60	901	899	60	1802	1798	120	2703	2697	180	3604	3596	240
62	962	960	62	1924	1920	124	2886	2880	186	3848	3840	248
64	1025	1023	64	2050	2046	128	3075	3069	192	4100	4092	256
66	1090	1088	66	2180	2176	132	3270	3264	198	4360	4352	264
68	1157	1155	68	2314	2310	136	3471	3465	204	4628	4620	272
70	1226	1224	70	2452	2448	140	3678	3672	210	4904	4896	280
72	1297	1295	72	2594	2590	144	3891	3885	216	5188	5180	288

conclusion: k parameter is a linear common factor of a,b,c

-Patrick Stoltz Dépot INPI N° 343319

patrick.stoltz@cegetel.net