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.

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.

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.

The first few such diagonals are the 8th, 12th, 26th, 35th, 38th, 53rd, 66th, 73rd, 77th, 90th, 121st, and 126th. These correspond to the composite numbers 15, 21, 39, 51, 55, 75, 91, 99, 105, 121, 159, and 165 respectively.

The Second Borve Conjecture states that there is an infinite number of zero-only diagonals.

Some elements in the first column belong to both a zero-only antidiagonal and a zero-only diagonal. These include the 8th and 12th such elements, namely B(8,1) and B(12,1). These correspond to the composite numbers 15 and 21 respectively.

The first few such elements are the 8th, 12th, 35th, and 73rd. These correspond to the composite numbers 15, 21, 51, and 99 respectively.

The Third Borve Conjecture states that there is an infinite number of column-1 elements which belong to both a zero-only antidiagonal and a zero-only diagonal.

Sequences

The three conjectures are equivalent to the conjectures that the following sequences of integers are infinite:

 Composites for which the row includes the leftmost element of a zero-only antidiagonal:: 9, 15, 21, 35, 45, 51, 65, 69, 95, 99, 119, 129, 135, 141, 155, 161, 189,...

 Composites for which the row includes the leftmost element of a zero-only diagonal: 15, 21, 39, 51, 55, 75, 91, 99, 105, 121, 159, 165,...

 Composites for which the row includes the leftmost element of both a zero-only antidiagonal and a zero-only diagonal: 15, 21, 51, 99,...

Other zero-only straight alignments

Also apparent in the table are zero-only 'knight's move' alignments.

For example, one such alignment begins at the first-column element in the fourth row (for the composite number 9) and contains all the cells that can be reached through successive 'moves' of 'two squares to the right and one square up'.

A similar zero-only alignment also occurs for the composite number 15.

I propose to call these alignments U,2,1-alignments where the 'U' stands for 'UP', in accordance with the fact that the alignment points up as it goes to the right. The '2,1' stands for '2 steps right, 1 step up'.

Alignments starting in the first column and made by 'moves' of '2 steps right, 1 step down' are termed D,2,1-alignments, where 'D' stands for 'DOWN'.

The first two zero-only D,2,1-alignments are for the composites 15 and 21.

Further Conjectures

A composite for which the U,2,1-alignment is zero-only is termed a U,2,1-composite.

A composite for which the D,2,1-alignment is zero-only is termed a D,2,1-composite.

A composite for which both the U,2,1-alignment and the D,2,1-alignment are zero-only is termed a V,2,1-composite (because the two alignments form a 'V'). The first V,2,1-composite is 15.

In this notation, the First, Second, and Third Borve Conjectures can be described respectively as:

the Borve U,1,1 Conjecture, namely that there is an infinite number of U,1,1-composites;

the Borve D,1,1 Conjecture, namely that there is an infinite number of D,1,1-composites; and

the Borve V,1,1 Conjecture, namely that there is an infinite number of V,1,1-composites.

Regarding the 'knight's move' alignments described above, it is conjectured that:

there is an infinite number of U,2,1-composites (the Borve U,2,1 Conjecture);

there is an infinite number of D,2,1-composites (the Borve D,2,1 Conjecture); and

there is an infinite number of V,2,1-composites (the Borve V,2,1 Conjecture).

In general, positive integers r, u, and d (for the number of steps 'RIGHT', 'UP', and 'DOWN' respectively) can be chosen so as to yield an infinite number of conjectures. These take the following form:

that there is an infinite number of U,r,u-composites (for specified r and u);

that there is an infinite number of D,r,d-composites (for specified r and d);

that there is an infinite number of V,r,v-composites (this is the conjecture that both the U,r,u Conjecture and the D,r,d Conjecture are true, for specified composites r and v, where v = u = d).

Numbers which are U,r,1-composite for all positive integers 1 <= r <= k are termed k-U-fan-composite (the word 'fan' is used because the alignments 'fan out' in one direction). The first numbers which are both U,1,1-composite and U-2,1-composite, and which are therefore 2-U-fan composite, are 9 and 15.

Similarly, numbers which are D,r,1-composite for all positive integers 1 <= r <= k are termed k-D-fan-composite. The first 2-D-fan composite is 15.

We observe that 15 is the first number which is both 2-U-fan composite and 2-D-fan composite (in other words, which is both V,1,1-composite and V,2,1-composite). In general, we describe as k-star-composite any number that is both k-U-fan composite and k-D-fan composite. The first 2-star composite is 15.

It is conjectured that:

for any positive integer k, there is at least one k-U-fan composite (the Borve k-U-fan Conjecture);

for any positive integer k, there is at least one k-D-fan composite (the Borve k-D-fan Conjecture);

for any positive integer k, there is at least one k-D-star composite (the Borve k-star Conjecture).

The much stronger conjectures are also made that:

for any positive integer k, there is an infinite number of k-U-fan composites (the Strong Borve k-U-fan Conjecture);

for any positive integer k, there is an infinite number of k-U-fan composites (the Strong Borve k-D-fan Conjecture);

for any positive integer k, there is an infinite number of k-U-fan composites (the Strong Borve k-star Conjecture).

Further conjectures can be made which refer to U,r,p-compositeness and D,r,p-compositeness where p > 1.

One especially general conjecture is that

for any composite c(m), there is an infinite number of pairs of the form (r,s) (where r is positive and s is positive or negative) such that c(m) is U,r,s-composite (when s is negative, U,r,s-compositeness being another way of denoting D,r,-s-compositeness); or, put graphically, that an infinite number of zero-only lines can be drawn from the first element of each row. Since for a given m the number of zero-only lines that can be drawn upwards is finite, this is equivalent to the conjecture that from the first element of the mth row an infinite number of zero-only lines can be drawn downwards.

The Exploring Primeness Project will report on further investigations in due course.

We welcome correspondence. We would be especially interested if people could email us (primeness (at) borve (dot) org) who are aware of any previous discussion of the prime-composite array. Since at present we do not benefit from access to high-powered computing facilities, we would also be grateful for assistance in extending both the array and the sequences which are conjectured to be infinite.

Footnote

1) The Conjectures are named after the village on the Atlantic coast of the Isle of Lewis, Scotland, which inspired their formulation.

Copyright, Neil Fernandez 2001, 2007.

Last modified: 14 October 2007.