Cutting by n an odd square

"Slicing" by n:

Calculation of c (fc(n)

Calculation of a (fa(n)

factorize

verification

C² – A² = B²

(c+a)(c-a)

B²=(fc(n)+fa(n))(fc(n)-fa(n))

B² = ((n²-1)/2 +1 + (n²-1)/2 )((n²-1)/2 +1 - (n²-1)/2 )

B² = ((n²-1)/2 +2/2 + (n²-1)/2 )((n²-1)/2 +2/2 - (n²-1)/2 )

(n²-1)/2 + (n²-1)/2 = 2(n²-1)/2 , Elimination de (n²-1)/2

B² = (2(n²-1)/2 +2/2 )(+2/2 )

B² = (2(n²-1)+2/2 )

B² = (2((n²-1)+1)/2 )

B² = (n²-1)+1

B²=n²

Matrice

N	C	B (or A)	A (or B)	C²	B²	A²	B²+A²
3	5	4	3	25	16	9	25
5	13	12	5	169	144	25	169
7	25	24	7	625	576	49	625
9	41	40	9	1681	1600	81	1681
11	61	60	11	3721	3600	121	3721
13	85	84	13	7225	7056	169	7225
15	113	112	15	12769	12544	225	12769
17	145	144	17	21025	20736	289	21025
19	181	180	19	32761	32400	361	32761
21	221	220	21	48841	48400	441	48841
23	265	264	23	70225	69696	529	70225
25	313	312	25	97969	97344	625	97969
27	365	364	27	133225	132496	729	133225
29	421	420	29	177241	176400	841	177241
31	481	480	31	231361	230400	961	231361
33	545	544	33	297025	295936	1089	297025
35	613	612	35	375769	374544	1225	375769
37	685	684	37	469225	467856	1369	469225
39	761	760	39	579121	577600	1521	579121
41	841	840	41	707281	705600	1681	707281
43	925	924	43	855625	853776	1849	855625
45	1013	1012	45	1026169	1024144	2025	1026169
47	1105	1104	47	1221025	1218816	2209	1221025
49	1201	1200	49	1442401	1440000	2401	1442401
51	1301	1300	51	1692601	1690000	2601	1692601
53	1405	1404	53	1974025	1971216	2809	1974025
55	1513	1512	55	2289169	2286144	3025	2289169

Slicing an odd square by n coefficiented by k

In the previous 2 topology, I have "to stretch" a strip of thickness 1 (or 1+1/2 in the even topology), i introduce a coefficient k allowing to widen this strip.

K represent the thickness of the strip (or c-a < > 1)

Sample: N=5, k=2

Calculation de b

calculation of c ( fc(n,k) )

calculation of a ( fa(n,k) )

factorisation

distribution

factorize k on numérator

factorize k

factorisation

distribution

factorisation k

verification : C² – A² = B² ; = (C+A)(C-A)

B²=((k(n² – 1)/2 + k)+(k(n²-1)/2 ))((k(n² – 1)/2 + k)-(k(n²-1)/2 ))

B²=((2k(n² – 1)/2 +k)(k)

B²=(k(n²-1)+k)(k)

B²=k²(n²-1)+k²

B²=k²((n²-1+1)

B²=k²n²

B=kn

	k=1			k=2			k=3			k=4
N	C	A	B	C	A	B	C	A	B	C	A	B
3	5	4	3	10	8	6	15	12	9	20	16	12
5	13	12	5	26	24	10	39	36	15	52	48	20
7	25	24	7	50	48	14	75	72	21	100	96	28
9	41	40	9	82	80	18	123	120	27	164	160	36
11	61	60	11	122	120	22	183	180	33	244	240	44
13	85	84	13	170	168	26	255	252	39	340	336	52
15	113	112	15	226	224	30	339	336	45	452	448	60
17	145	144	17	290	288	34	435	432	51	580	576	68
19	181	180	19	362	360	38	543	540	57	724	720	76
21	221	220	21	442	440	42	663	660	63	884	880	84
23	265	264	23	530	528	46	795	792	69	1060	1056	92
25	313	312	25	626	624	50	939	936	75	1252	1248	100
27	365	364	27	730	728	54	1095	1092	81	1460	1456	108
29	421	420	29	842	840	58	1263	1260	87	1684	1680	116
31	481	480	31	962	960	62	1443	1440	93	1924	1920	124
33	545	544	33	1090	1088	66	1635	1632	99	2180	2176	132
35	613	612	35	1226	1224	70	1839	1836	105	2452	2448	140
37	685	684	37	1370	1368	74	2055	2052	111	2740	2736	148
39	761	760	39	1522	1520	78	2283	2280	117	3044	3040	156
41	841	840	41	1682	1680	82	2523	2520	123	3364	3360	164
43	925	924	43	1850	1848	86	2775	2772	129	3700	3696	172
45	1013	1012	45	2026	2024	90	3039	3036	135	4052	4048	180
47	1105	1104	47	2210	2208	94	3315	3312	141	4420	4416	188
49	1201	1200	49	2402	2400	98	3603	3600	147	4804	4800	196
51	1301	1300	51	2602	2600	102	3903	3900	153	5204	5200	204
53	1405	1404	53	2810	2808	106	4215	4212	159	5620	5616	212
55	1513	1512	55	3026	3024	110	4539	4536	165	6052	6048	220
57	1625	1624	57	3250	3248	114	4875	4872	171	6500	6496	228
59	1741	1740	59	3482	3480	118	5223	5220	177	6964	6960	236
61	1861	1860	61	3722	3720	122	5583	5580	183	7444	7440	244
63	1985	1984	63	3970	3968	126	5955	5952	189	7940	7936	252
65	2113	2112	65	4226	4224	130	6339	6336	195	8452	8448	260
67	2245	2244	67	4490	4488	134	6735	6732	201	8980	8976	268
69	2381	2380	69	4762	4760	138	7143	7140	207	9524	9520	276
71	2521	2520	71	5042	5040	142	7563	7560	213	10084	10080	284

conclusion: k parameter is also a linear common factor of a,b,c in odd topologic squares

Patrick Stoltz le 19/02/2009 – dépôt INPI n°: 343319 -

patrick.stoltz@cegetel.net