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 6y1 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):
(2x1)^2 + 2(n)(2x1) (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 
A^{1}^{} 

B 

C 

A 

D 

D 

A 

C 

B 

A 


x 
2x1 
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, (2x1)^2 + 2(n)(2x1). 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 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 
6y1 
4 
0,3,6,9 
A 
6y+1 
5 
2,7 
C 
6y1 
7 
0,3,6,9 
A 
6y+1 
8 
1,8 
B 
6y1 
**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
6y1. 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, 6y1 and 6y+1:
Setting 6y1 = (2x1)^2 + 2(n)(2x1) and solving for the integer solutions gives:
1(a) n = 3m1 + 1, x =
1(b) n = 3m1 + 2, x =
Similarly, the integer solutions for 6y+1 = (2x1)^2 + 2(n)(2x1) gives:
2(a)
2(b)
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 =
1(a) m_{1} 
m_{2} 
n=3m_{1}+1 
x=3m_{2} 
1(a) y 
Form: 6y1 
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 =
1(b) m_{1} 
m_{2} 
n=3m_{1}+2 
x=3m_{2}+1 
1(b) y 
Form: 6y1 
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 =
2(a) m_{1} 
m_{2} 
n=3m_{1} 
x=3m_{2} 
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 =
2(b) m_{1} 
m_{2} 
n=3m_{1} 
x=3m_{2}+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 
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 6y1 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 6y1 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 
6y1 
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 
6y1 
2 
1 
1 
1(a) 
D 
6 
77 
6y1 
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 
6y1 
5 
2 
1 
1(a) 
C 
10 
115 
6y+1 
7 
3 
1 
2(a) 
A 
11 
119* 
6y1 
2 
1 
1 
1(b) 
D 
12 
121* 
6y+1 
4 
0 
2 
2(a) 
A 
13 
125 
6y1 
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.
From a general equation for all odd composite numbers, integer solutions are obtained for those odd composite numbers of the Forms 6y1 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
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 
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: (2x1)^2 + 2(n)(2x1) with 2x1 as the odd numbers.