The Odd Composite Numbers

 The Odd Composite Numbers

Richard G. Lanzara

From a general equation for all odd composite numbers, integer solutions are obtained for those odd composite numbers of the Forms 6y-1 and 6y+1. Relative to finding twin primes, the twin odd composites were found where there would normally be the twin primes between those odd composites divisible by 3. The first example of this occurrence is for the twin odd composite numbers 119,121. Some of the patterns and symmetries are described for the odd composite numbers including their relationships to the digit sums or mod(9) values. 

All odd composite numbers can be represented by (see: The Appendix for how this equation was derived):

                                (2x-1)^2 + 2(n)(2x-1)          (1)

For example: 63 = 7^2 + (2)(7). Equation (1) is a useful formula that sets apart all the odd composite numbers from the prime numbers. The beauty of this approach becomes evident if we examine a table of the odd composite numbers generated by eq. (1) (Table I).

Table I: The Odd Composite Numbers

Cols		A1		B		C		A		D		D		A		C		B		A
x	2x-1	n=0	ds	n=1	ds	n=2	ds	n=3	ds	n=4	ds	n=5	ds	n=6	ds	n=7	ds	n=8	ds	n=9	ds
1	1*	1	1	3	3	5	5	7	7	9	9	11	2	13	4	15	6	17	8	19	1
2	3	9	9	15	6	21	3	27	9	33	6	39	3	45	9	51	6	57	3	63	9
3	5	25	7	35	8	45	9	55	1	65	2	75	3	85	4	95	5	105	6	115	7
4	7	49	4	63	9	77	5	91	1	105	6	119	2	133	7	147	3	161	8	175	4
5	9	81	9	99	9	117	9	135	9	153	9	171	9	189	9	207	9	225	9	243	9
6	11	121	4	143	8	165	3	187	7	209	2	231	6	253	1	275	5	297	9	319	4
7	13	169	7	195	6	221	5	247	4	273	3	299	2	325	1	351	9	377	8	403	7
8	15	225	9	255	3	285	6	315	9	345	3	375	6	405	9	435	3	465	6	495	9
9	17	289	1	323	8	357	6	391	4	425	2	459	9	493	7	527	5	561	3	595	1
10	19	361	1	399	3	437	5	475	7	513	9	551	2	589	4	627	6	665	8	703	1

1) The first row contains all the odd numbers (including the primes) with x=1 in the equation, (2x-1)^2 + 2(n)(2x-1). After the first row, which contains all the odd numbers, the other rows contain all the odd composite numbers for x>1.

Each of the columns (A,B,C,A,D,D,A,C,B,A,…) continuously repeat and have an internal bilateral symmetry. They also have other repeating properties such as repeating digit sums (“ds” columns to the right for columns A, B, etc.) The ds values reading down column A are 1,9,7,4,9,4,7,9,1,…, and column B values are 3,6,8,9,9,8,6,3,8,3,…, etc., which repeat after 9 (mod(9)). (The digit sum is the result of repeatedly computing the sum of the digits until a single digit answer is obtained. The digit sum of a number n is often denoted as DigitSum(n). This is also the same as the number mod(9).)  

 Table I shows that the digit sums of 3,6 and 9 are all the odd composite numbers divisible by 3; whereas the digit sums of 1,4,7 are found only in the A columns, while the digit sum of 2 is only found in the D columns, and the digit sum of 5 is only found in the C columns, and the digit sum of 8 is only found in the B columns. This is summarized in Table II. 

Table II: Comparisons of the Values for n in Equation (1) and the Digit Sums

Digit sums (ds)	Values of n**	Columns A to D	Forms
1	0,3,6,9	A	6y+1
2	4,5	D	6y-1
4	0,3,6,9	A	6y+1
5	2,7	C	6y-1
7	0,3,6,9	A	6y+1
8	1,8	B	6y-1

**These are the smallest values for n in Equation (1). There is a bigger pattern, which is n plus a multiple of 9 (i.e. (n+9k), k=1, 2, 3,…) that links n values with their respective columns (such as 2+9k and 7+9k with the “C” Column and 4+9k and 5+9k with the “D’ Column, etc.).

Table II also contains the ‘Forms’ for the odd composite numbers found only in specific columns A to D. In general, the A columns contain only the Form 6y+1 and the B, C and D columns contain only the Form 6y-1. Numbers of the Form 6y+3 are not considered since they are always divisible by 3.

The beauty of this approach becomes evident if we examine the two basic forms for the odd composite numbers not divisible by 3, 6y-1 and 6y+1:

Setting 6y-1 = (2x-1)^2 + 2(n)(2x-1) and solving for the integer solutions gives:

1(a)  n = 3m1 + 1,  x = 3m2, y = 6(m2)^2 + 6m2m1 - m1

1(b)  n = 3m1 + 2,  x = 3m2 + 1, y = 6(m2)^2 + 6m2m1 + 6m2 + m1 + 1

Similarly, the integer solutions for 6y+1 = (2x-1)^2 + 2(n)(2x-1) gives:

2(a)  n = 3m1,  x = 3m2, y = 6(m2)^2 + 6m2m1 - 2m2 - m1 

2(b)  n = 3m1,  x = 3m2 + 1, y = 6(m2)^2 + 6m2m1 + 2m2 + m1 

These four equations for the integer solutions include all solutions for the odd composite numbers not divisible by 3.

1(a)  n = 3m1 + 1,  x = 3m2, y = 6(m2)^2 + 6m2m1 - m1 with a solutions table:

1(a) m1	m2	n=3m1+1	x=3m2	1(a) y	Form: 6y-1	ds	Column
0	1	1	3	6	35	8	B
	2	1	6	24	143	8	B
	3	1	9	54	323	8	B
	4	1	12	96	575	8	B
1	1	4	3	11	65	2	D
	2	4	6	35	209	2	D
	3	4	9	71	425	2	D
	4	4	12	119	713	2	D
2	1	7	3	16	95	5	C
	2	7	6	46	275	5	C
	3	7	9	88	527	5	C
	4	7	12	142	851	5	C

1(b)  n = 3m1 + 2,  x = 3m2 + 1, y = 6(m2)^2 + 6m2m1 + 6m2 + m1 + 1 with a solutions table:

1(b) m1	m2	n=3m1+2	x=3m2+1	1(b) y	Form: 6y-1	ds	Column
0	1	2	4	13	77	5	C
	2	2	7	37	221	5	C
	3	2	10	73	437	5	C
	4	2	13	121	725	5	C
1	1	5	4	20*	119	2	D
	2	5	7	50	299	2	D
	3	5	10	92	551	2	D
	4	5	13	146	875	2	D
2	2	8	7	63	377	8	B
	3	8	10	111*	665	8	B
	4	8	13	171	1025	8	B

 

2(a)  n = 3m1,  x = 3m2, y = 6(m2)^2 + 6m2m1 - 2m2 - m1 with a solutions table:

2(a) m1	m2	n=3m1	x=3m2	2(a) y	Form: 6y+1	ds	Column
0	1	0	3	4	25	7	A
	2	0	6	20*	121	4	A
	3	0	9	48	289	1	A
	4	0	12	88	529	7	A
1	2	3	6	31	187	7	A
	3	3	9	65	391	4	A
	4	3	12	111*	667	1	A
2	2	6	6	42	253	1	A
	3	6	9	82	493	7	A
	4	6	12	134	805	4	A

*When the y values are equal, then the odd composites will be twins (eg. 1b) y=20, 119 and 2a) y=20, 121).

2(b)  n = 3m1,  x = 3m2 + 1, y = 6(m2)^2 + 6m2m1 + 2m2 + m1 with a solutions table:

2(b) m1	m2	n=3m1	x=3m2+1	2(b) y	Form: 6y+1	ds	Column
0	1	0	4	8	49	4	A
	2	0	7	28	169	7	A
	3	0	10	60	361	1	A
	4	0	13	104	625	4	A
1	2	3	7	41	247	4	A
	3	3	10	79	475	7	A
	4	3	13	129	775	1	A
2	2	6	7	54	325	1	A
	3	6	10	98	589	4	A
	4	6	13	154	925	7	A

 It is interesting to consider where twin odd composites appear relative to the twin primes. The twin primes are bounded by two odd composite numbers (ignoring 3,5 and 5,7, which are really a prime triplet). So for example, 11,13 is bounded by 9,15 and 17,19 is bounded by 15,21. So checking many of these twin primes, one can propose that the twin primes are bounded only by the odd composite numbers that have the digit sums (ds) of 3, 6, or 9. This means that the pairs of odd composite numbers that bound and contain the twin primes are of the following pairs of digit sums (3,9), (6,3) or (9,6) and not of the forms (1,7), (2,8), (4,1), (5,2), (7,4) or (8,5).

This may be proved by considering that by adding 2 then 4 to determine the digit sums of the twin prime pair would give digit sums incompatible with one of the numbers being a prime. For example, (1+2=3, which is a number divisible by 3, and 2+4=6, which is also a number divisible by 3). Similarly each one of the digit sums that are not allowed (those with ds values (1,7), (2,8), (4,1), (5,2), (7,4) or (8,5)) would generate at least one number with a digit sum of 3, 6 or 9, which are odd composite numbers divisible by 3 and therefore, will never be a prime.

Therefore, the twin primes are found only between the odd composite numbers with digit sums of (3,9), (6,3) or (9,6) exclusively.  This means that the only possible digit sums for the twin primes or odd composites is (8,1)(5,7)(2,4). This implies that for the odd composite numbers the only possible pairings of columns are (B,A)(C,A)(D,A), which is consistent with the observation that the Form 6y-1 occurs only in columns B,C or D and Form 6y+1 occurs only in the A columns.

Table III: Exploring the Space of the Odd Composites of the Forms 6y-1 and 6y+1.

	Odd Composite	Form	Digit sum (ds)	m1	m2	Integer Equation	Column
1	25	6y+1	7	0	1	2(a)	A
2	35	6y-1	8	0	1	1(a)	B
3	49	6y+1	4	0	1	2(b)	A
4	55	6y+1	1	1	1	2(a)	A
5	65	6y-1	2	1	1	1(a)	D
6	77	6y-1	5	0	1	1(b)	C
7	85	6y+1	4	2	1	2(a)	A
8	91	6y+1	1	1	1	2(b)	A
9	95	6y-1	5	2	1	1(a)	C
10	115	6y+1	7	3	1	2(a)	A
11	119*	6y-1	2	1	1	1(b)	D
12	121*	6y+1	4	0	2	2(a)	A
13	125	6y-1	8	3	1	1(a)	B

*The first occurrence of twin odd composites where the twin primes would normally occur. Those composites that have ds=3, 6 or 9 are omitted from this table.

 

Table IV: The Twin Primes or Twin Odd Composite Numbers

	Twin Odd Composites	ds	Twin Primes*	ds
1	119,121	(2,4)	11,13	(2,4)
2	143,145	(8,1)	17,19	(8,1)
3	185,187	(5,7)	29,31	(2,4)
4	203,205	(5,7)	41,43	(5,7)
5	215,217	(8,1)	59,61	(5,7)
6	245,247	(2,4)	71,73	(8,1)
7	287,289	(8,1)	101,103	(2,4)
8	299,301	(2,4)	107,109	(8,1)
9	323,325	(8,1)	137,139	(2,4)
10	341,343	(8,1)	149,151	(5,7)
11	413,415	(8,1)	179,181	(8,1)
12	425,427	(2,4)	191,193	(2,4)
13	473,475	(5,7)	197,199	(8,1)
14	515,517	(2,4)	227,229	(2,4)
15	527,529	(5,7)	239,241	(5,7)
16	533,535	(2,4)	269,271	(8,1)
17	551,553	(2,4)	281,283	(2,4)
18	581,583	(5,7)	311,313	(5,7)
19	623,625	(5,7)	347,349	(5,7)
20	635,637	(8,1)	419,421	(5,7)

*The twin primes, (3,5) and (5,7) are not included in this table.

 Conclusion

From a general equation for all odd composite numbers, integer solutions are obtained for those odd composite numbers of the Forms 6y-1 and 6y+1. Relative to finding twin primes, the twin odd composites were found where there would normally be the twin primes between those odd composites divisible by 3. The first example of this occurrence is for the twin odd composite numbers 119,121. It is interesting how few there are of the first twin odd composites using our definition eliminating those divisible by 3. Some of the patterns and symmetries are also intriguing for the odd composite numbers including their relationships to the digit sums or mod(9) values.

Appendix

 Examining a simple multiplication table,

Table I: Simple Multiplication

	x = 2	3	4	5	6	7	8	9	10
2	4
3	6	9
4	8	12	16
5	10	15	20	25
6	12	18	24	30	36
7	14	21	28	35	42	49
8	16	24	32	40	48	56	64
9	18	27	36	45	54	63	72	81
10	20	30	40	50	60	70	80	90	100

x^2 + nx, with n,x positive integers, is a general equation for the diagonal numbers in the simple multiplication table

For example: 35 = 5^2 + (2)(5); 54 = 6^2 + (3)(6); and 63 = 7^2 + (2)(7). Where “n” is the slope, which is the number of rows beneath the square number. 

To express only the odd composite numbers, we need to modify the general equation above to: (2x-1)^2 + 2(n)(2x-1) with 2x-1 as the odd numbers.