Pythagorean Triples

A traditional method of finding triple sets of numbers that work so that a^2 + b^2 = c^2 starts by choosing any p and a corresponding q that is, at most, p-1. Then, 2pq, p^2-q^2, and p^2+q^2 provide 3 numbers that work. The actual squared values would be:

(2pq)^2 + (p^2-q^2)^2 = (p^2+q^2)^2

Either the first or second number may be larger.

From Dr. Math:

Thus the formulas that give all Pythagorean triples are these:

   a = d*(m^2 - n^2),
   b = 2*d*m*n,
   c = d*(m^2 + n^2),

where d is any positive integer, m > n > 0 are integers of opposite 
parity and relatively prime.

. . . If you are given a triple a, b, and c, you can find d, m, and n in the
following way. First of all, d = GCD(a,c). Then m = Sqrt[(c+a)/(2*d)] 
and n = Sqrt[(c-a)/(2*d)].

Now consider

x = 2pq - (p^2 - q^2) (absolute value)

Since EITHER 2pq + x = b

OR (p^2 - q^2) + x = b [NOT BOTH]

The other one of the two equals a.

Thus 2pq + x + (p^2 - q^2) = 2b

And for a < b < c:

b = pq + x/2 + (p^2 - q^2)/2

a = pq - x/2 + (p^2 - q^2)/2

c = p^2 + q^2

u shells = c - b and u value = a^2

a + x + u = c for both integer # of shells and summed shell values

CHART OF VALUES, a < b < c:

p	q			c	c	x	b	a	u		u value	x value
p	q	p^2-q^2	2pq	p^2 +q^2	2pq + (p-q)^2	absolute value │‌2pq-(p^2-q^2)‌│=x	pq+(x/2) +(p^2-q^2)/2 =b	pq-(x/2) +(p^2-q^2)/2 =a		(c^2-b^2) /a^2	c^2 - b^2	c^2 - (c^2-b^2) - a^2	a^n + x + u = c^n	u value = a^n	a^n + x value = b^n
2	1	3	4	5	5	1	4	3	1	1	9	7	TRUE	TRUE	TRUE
3	1	8	6	10	10	2	8	6	2	1	36	28	TRUE	TRUE	TRUE
3	2	5	12	13	13	7	12	5	1	1	25	119	TRUE	TRUE	TRUE
4	1	15	8	17	17	7	15	8	2	1	64	161	TRUE	TRUE	TRUE
4	2	12	16	20	20	4	16	12	4	1	144	112	TRUE	TRUE	TRUE
4	3	7	24	25	25	17	24	7	1	1	49	527	TRUE	TRUE	TRUE
5	1	24	10	26	26	14	24	10	2	1	100	476	TRUE	TRUE	TRUE
5	2	21	20	29	29	1	21	20	8	1	400	41	TRUE	TRUE	TRUE
5	3	16	30	34	34	14	30	16	4	1	256	644	TRUE	TRUE	TRUE
5	4	9	40	41	41	31	40	9	1	1	81	1519	TRUE	TRUE	TRUE
6	1	35	12	37	37	23	35	12	2	1	144	1081	TRUE	TRUE	TRUE
6	2	32	24	40	40	8	32	24	8	1	576	448	TRUE	TRUE	TRUE
6	3	27	36	45	45	9	36	27	9	1	729	567	TRUE	TRUE	TRUE
6	4	20	48	52	52	28	48	20	4	1	400	1904	TRUE	TRUE	TRUE
6	5	11	60	61	61	49	60	11	1	1	121	3479	TRUE	TRUE	TRUE
7	1	48	14	50	50	34	48	14	2	1	196	2108	TRUE	TRUE	TRUE
7	2	45	28	53	53	17	45	28	8	1	784	1241	TRUE	TRUE	TRUE
7	3	40	42	58	58	2	42	40	16	1	1600	164	TRUE	TRUE	TRUE
7	4	33	56	65	65	23	56	33	9	1	1089	2047	TRUE	TRUE	TRUE
7	5	24	70	74	74	46	70	24	4	1	576	4324	TRUE	TRUE	TRUE
7	6	13	84	85	85	71	84	13	1	1	169	6887	TRUE	TRUE	TRUE
8	1	63	16	65	65	47	63	16	2	1	256	3713	TRUE	TRUE	TRUE
8	2	60	32	68	68	28	60	32	8	1	1024	2576	TRUE	TRUE	TRUE
8	3	55	48	73	73	7	55	48	18	1	2304	721	TRUE	TRUE	TRUE
8	4	48	64	80	80	16	64	48	16	1	2304	1792	TRUE	TRUE	TRUE
8	5	39	80	89	89	41	80	39	9	1	1521	4879	TRUE	TRUE	TRUE
8	6	28	96	100	100	68	96	28	4	1	784	8432	TRUE	TRUE	TRUE
8	7	15	112	113	113	97	112	15	1	1	225	12319	TRUE	TRUE	TRUE
9	1	80	18	82	82	62	80	18	2	1	324	6076	TRUE	TRUE	TRUE
9	2	77	36	85	85	41	77	36	8	1	1296	4633	TRUE	TRUE	TRUE
9	3	72	54	90	90	18	72	54	18	1	2916	2268	TRUE	TRUE	TRUE
9	4	65	72	97	97	7	72	65	25	1	4225	959	TRUE	TRUE	TRUE
9	5	56	90	106	106	34	90	56	16	1	3136	4964	TRUE	TRUE	TRUE
9	6	45	108	117	117	63	108	45	9	1	2025	9639	TRUE	TRUE	TRUE
9	7	32	126	130	130	94	126	32	4	1	1024	14852	TRUE	TRUE	TRUE
9	8	17	144	145	145	127	144	17	1	1	289	20447	TRUE	TRUE	TRUE
10	1	99	20	101	101	79	99	20	2	1	400	9401	TRUE	TRUE	TRUE
10	2	96	40	104	104	56	96	40	8	1	1600	7616	TRUE	TRUE	TRUE
10	3	91	60	109	109	31	91	60	18	1	3600	4681	TRUE	TRUE	TRUE
10	4	84	80	116	116	4	84	80	32	1	6400	656	TRUE	TRUE	TRUE
10	5	75	100	125	125	25	100	75	25	1	5625	4375	TRUE	TRUE	TRUE
10	6	64	120	136	136	56	120	64	16	1	4096	10304	TRUE	TRUE	TRUE
10	7	51	140	149	149	89	140	51	9	1	2601	16999	TRUE	TRUE	TRUE
10	8	36	160	164	164	124	160	36	4	1	1296	24304	TRUE	TRUE	TRUE
10	9	19	180	181	181	161	180	19	1	1	361	32039	TRUE	TRUE	TRUE
11	1	120	22	122	122	98	120	22	2	1	484	13916	TRUE	TRUE	TRUE
11	2	117	44	125	125	73	117	44	8	1	1936	11753	TRUE	TRUE	TRUE
11	3	112	66	130	130	46	112	66	18	1	4356	8188	TRUE	TRUE	TRUE
11	4	105	88	137	137	17	105	88	32	1	7744	3281	TRUE	TRUE	TRUE
11	5	96	110	146	146	14	110	96	36	1	9216	2884	TRUE	TRUE	TRUE
11	6	85	132	157	157	47	132	85	25	1	7225	10199	TRUE	TRUE	TRUE
11	7	72	154	170	170	82	154	72	16	1	5184	18532	TRUE	TRUE	TRUE
11	8	57	176	185	185	119	176	57	9	1	3249	27727	TRUE	TRUE	TRUE
11	9	40	198	202	202	158	198	40	4	1	1600	37604	TRUE	TRUE	TRUE
11	10	21	220	221	221	199	220	21	1	1	441	47959	TRUE	TRUE	TRUE
12	1	143	24	145	145	119	143	24	2	1	576	19873	TRUE	TRUE	TRUE
12	2	140	48