The Exploring Primeness Project
Neil Fernandez
The prime-composite array, B(m,n), and the Borve conjectures
The prime-composite array
Let c(m) be the mth composite and p(n) be the nth prime.
The prime-composite array, B, is defined such that each element B(m,n) is the highest power of p(n) that is contained within c(m).
Thus each composite has its own row, consisting of the indices of its prime factors. For example, the 10th composite is 18, and 18 = 21 * 32 * 50 * 70 * 110 * ..., so the 10th row reads: 1, 2, 0, 0, 0, ...
Similarly, B(6,2) = 1, because c(6) = 12, p(2) = 3, and the highest power of 3 contained within 12 is 31 = 3. And B(34,3) = 2, because c(34) = 50, p(3) = 5, and the highest power of 5 contained within 50 is 52 = 25.
Here is the top left corner of the array. For ease of reference the primes and composites themselves are also shown, in parentheses.
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
… |
n |
||
(2) |
(3) |
(5) |
(7) |
(11) |
(13) |
(17) |
(19) |
(23) |
(29) |
(31) |
(37) |
(41) |
(43) |
… |
(p(n)) |
||
1 |
(4) |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
2 |
(6) |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
3 |
(8) |
3 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
4 |
(9) |
0 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
5 |
(10) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
6 |
(12) |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
7 |
(14) |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
8 |
(15) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
9 |
(16) |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
10 |
(18) |
1 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
11 |
(20) |
2 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
12 |
(21) |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
13 |
(22) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
14 |
(24) |
3 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
… |
… |
||||||||||||||||
m |
(c(m)) |
The prime-composite array appears to be a very useful tool with which to study the distribution of prime factors and their powers among successive composite numbers.
In particular, there is great scope for finding and seeking different ways in which numbers in the table 'line up'. This may well be as great as the scope offered by the 'prime spirals' invented by Stanislaw Ulam in the 20th century.
In the present text, conjectures are made concerning straight alignments which start in the first column and contain only zeroes.
Antidiagonals and diagonals
The mth antidiagonal of the array consists of the m elements B(m,1), B(m-1,2), B(m-2,3),...,B(1,m).
Some antidiagonals contain only zeroes. These include the 4th, 8th, and 12th antidiagonals.
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
… |
n |
||
(2) |
(3) |
(5) |
(7) |
(11) |
(13) |
(17) |
(19) |
(23) |
(29) |
(31) |
(37) |
(41) |
(43) |
… |
(p(n)) |
||
1 |
(4) |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
2 |
(6) |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
3 |
(8) |
3 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
4 |
(9) |
0 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
5 |
(10) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
6 |
(12) |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
7 |
(14) |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
8 |
(15) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
9 |
(16) |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
10 |
(18) |
1 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
11 |
(20) |
2 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
12 |
(21) |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
13 |
(22) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
14 |
(24) |
3 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
… |
… |
||||||||||||||||
m |
(c(m)) |
The first few such antidiagonals are the 4th, 8th, 12th, 23rd, 30th, 35th, 46th, 49th, 70th, 73rd, 88th, 97th, 102nd, 106th, 118th, 123rd, and 146th. These correspond to the composite numbers 9, 15, 21, 35, 45, 51, 65, 69, 95, 99, 119, 129, 135, 141, 155, 161, and 189 respectively.
The First Borve Conjecture states that there is an infinite number of zero-only antidiagonals. [1].
Diagonals can also be specified, where the mth diagonal consists of the infinite number of elements B(m,1), B(m+1,2), B(m+2,3),...
Some diagonals contain only zeroes. These include the 8th, and 12th diagonals.
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
… |
n |
||
(2) |
(3) |
(5) |
(7) |
(11) |
(13) |
(17) |
(19) |
(23) |
(29) |
(31) |
(37) |
(41) |
(43) |
… |
(p(n)) |
||
1 |
(4) |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
2 |
(6) |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
3 |
(8) |
3 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
4 |
(9) |
0 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
5 |
(10) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
6 |
(12) |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
7 |
(14) |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
8 |
(15) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
9 |
(16) |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
10 |
(18) |
1 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
11 |
(20) |
2 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
12 |
(21) |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
13 |
(22) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
14 |
(24) |
3 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
||
… |