The Function x^2 + xy + y^2 – Part 7

7)      A number in the form of x² + xy + y² can have several different x and y solutions.

(x, y) = (y, x) = (-x, -y) = (-y, -x) = (x, -x – y) = (-x – y, x) = (-x, x + y) = (x + y, -x) = (-x – y, y) = (y, -x – y) = (x + y, -y) = (-y, x + y).  For example, if x = 2 and y = 3, then x² + xy + y² = 2² + 2∙3 + 3² = 19, so (2, 3) = 19.  Since -x – y = -5 and x + y = 5, 19 can also be obtained by (3, 2), (-2, -3), (-3, -2), (2, -5), (-5, 2), (-2, 5), (5, -2), (-5, 3), (3, -5), (5, -3) and (-3, 5).

	-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	9	10
10	100	91	84	79	76	75	76	79	84	91	100	111	124	139	156	175	196	219	244	271	300	10
9	91	81	73	67	63	61	61	63	67	73	81	91	103	117	133	151	171	193	217	243	271	9
8	84	73	64	57	52	49	48	49	52	57	64	73	84	97	112	129	148	169	192	217	244	8
7	79	67	57	49	43	39	37	37	39	43	49	57	67	79	93	109	127	147	169	193	219	7
6	76	63	52	43	36	31	28	27	28	31	36	43	52	63	76	91	108	127	148	171	196	6
5	75	61	49	39	31	25	21	19	19	21	25	31	39	49	61	75	91	109	129	151	175	5
4	76	61	48	37	28	21	16	13	12	13	16	21	28	37	48	61	76	93	112	133	156	4
3	79	63	49	37	27	19	13	9	7	7	9	13	19	27	37	49	63	79	97	117	139	3
2	84	67	52	39	28	19	12	7	4	3	4	7	12	19	28	39	52	67	84	103	124	2
1	91	73	57	43	31	21	13	7	3	1	1	3	7	13	21	31	43	57	73	91	111	1
0	100	81	64	49	36	25	16	9	4	1	0	1	4	9	16	25	36	49	64	81	100	0
-1	111	91	73	57	43	31	21	13	7	3	1	1	3	7	13	21	31	43	57	73	91	-1
-2	124	103	84	67	52	39	28	19	12	7	4	3	4	7	12	19	28	39	52	67	84	-2
-3	139	117	97	79	63	49	37	27	19	13	9	7	7	9	13	19	27	37	49	63	79	-3
-4	156	133	112	93	76	61	48	37	28	21	16	13	12	13	16	21	28	37	48	61	76	-4
-5	175	151	129	109	91	75	61	49	39	31	25	21	19	19	21	25	31	39	49	61	75	-5
-6	196	171	148	127	108	91	76	63	52	43	36	31	28	27	28	31	36	43	52	63	76	-6
-7	219	193	169	147	127	109	93	79	67	57	49	43	39	37	37	39	43	49	57	67	79	-7
-8	244	217	192	169	148	129	112	97	84	73	64	57	52	49	48	49	52	57	64	73	84	-8
-9	271	243	217	193	171	151	133	117	103	91	81	73	67	63	61	61	63	67	73	81	91	-9
-10	300	271	244	219	196	175	156	139	124	111	100	91	84	79	76	75	76	79	84	91	100	-10
	-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	9	10

Some of the equalities are due to the fact that x² + xy + y² is commutative, so (x, y) = (y, x) because x² + xy + y² = y² + yx + x².  Some of the other equalities are due to the fact that x² + xy + y² is symmetrical, so (x, y) = (-x, -y) because x² + xy + y² = (-x)² + (-x)(-y) + (-y)².  The rest of the equalities with -x – y (or x + y) term can also be proved by substitution.  For example, (x, -x – y) = x² + x(-x – y) + (-x – y)² = x² – x² – xy + x² + 2xy + y² = x² + xy + y² = (x, y).

To derive the equality (x, y) = (x, -x – y), we can let p = x² + xy + y² and solve the formula p = x² + x(y + k) + (y + k)² for k, knowing that one solution for k is k = 0, and recognizing that as a quadratic, k will likely have a second solution.  So solving, x² + xy + xk + y² + 2ky + k² – p = 0, or k² + (x + 2y)k + x² + xy + y² – p = 0.  Since p = x² + xy + y², then k² + (x + 2y)k = 0, or k(k + (x + 2y)) = 0.  So k = 0 (which we already knew) or k = -(x + 2y).  If k = -(x + 2y), then the other coordinate with x would be y + k = y + -(x + 2y) = -x – y.