The Function x^2 + xy + y^2 – Part 3

3)      Multiples of some numbers follow patterns in the table of x² + xy + y²

Multiples of numbers follow certain patterns.  Multiples of 2 only occur if x and y are also multiples of 2.

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

Multiples of three are arranged in diagonals on the table, which means that if x – y is a multiple of 3, then x² + xy + y² is also a multiple of 3.

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

This is because x² + xy + y² = (x – y)² + 3xy.  If x – y is a multiple of 3, then adding 3xy (another multiple of 3) would result in an answer that is also a multiple of 3.

Like 2, multiples of 5 only occur if x and y are also multiples of 5.

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

In fact, it appears that any multiple of a number in the form of 3n + 2 (where n is an integer) only occur if x and y are also a multiple of 3n + 2.  (In other words, 2, 5, 8, 11, etc.)

Multiples of 7 do not follow such a nice pattern, but the pattern repeats for every 7 rows and 7 columns (so if (1, 2) is a multiple of 7, then so is (1, 2 + 7) = (1, 9)).

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