Ratio Mathematica Volume 47, 2023

On the structure of composite odd integers
and prime numbers

Giuseppe Buffoni*

Abstract

With the expression “structure of an odd integer” we mean the set of
properties of the integer n which specifies the odd integer 2n + 1 and
brings about its behaviour. These properties of n, for both compos-
ite and prime numbers, are expounded in detail, together with their
geometrical implications. In this context, a set, in a two dimensional
space, where all the composite odd integers in [2a + 1, 2n + 1] are
localized, is illustrated.
Keywords: couples of divisors; no divisors.
2020 AMS subject classifications: 11A00, 11Y00.1

*CNR-IMATI “E. Magenes”, Milano, Italy; giuseppe.buffoni9av2@gmail.com
1Received on March 31, 2023. Accepted on June 1, 2023. Published on June 30, 2023.
DOI: 10.23755/rm.v41i0.967. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper
is published under the CC-BY licence agreement.

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G. Buffoni

1 Introduction
Let 2n + 1 be any composite odd integer, and 2i + 1, 2j + 1 a couple of its

divisors. In [1] it is pointed out that

2n + 1 = (2i + 1)(2j + 1) ⇐⇒ n = kij, (1)

where kij is the following symmetric form in i and j

kij = i + j + 2ij, (i, j) ∈ N. (2)

Let K = {kij|(i, j) ∈ N} ⊂ N. Since any odd integer is either a composite or a
prime number, it follows that

n ∈ K ⇐⇒ 2n + 1 is a composite number, n ∈ N \ K ⇐⇒ 2n + 1 is a
prime number.

Some properties of the matrix {kij} of finite order are reported in the appendix
1 of [1], where the relation n = kij has been used to identify composite odd
integers 2ν + 1, with 4 = k11 ≤ ν ≤ n, and having the same number of couples
of divisors. The distributions of odd integers ≤ 2n + 1 vs. the number of their
couple of divisors have been computed up to n ≃ 5 · 107.

The motivation for this work is just the reference [1], unique reference for this
paper, and very useful for the analysis of the problem under study.

In section 2 we enlighten the basic properties of n for both composite odd
numbers and primes. In section 3 their main geometrical implications are illus-
trated. In particular, the relation n = kij for composite odd integers suggested
to investigate the localization of the pairs (i, j) producing kij in a given interval.
Thus, a set Ω(a, n) of points (x, y) ∈ R2 containing all the points with integer
coordinates (i, j) such that a ≤ kij ≤ n is defined.

2 The basic properties of n specifying the behaviour
of 2n + 1

For a pair (i, j), identifying any possible couple of divisors of an odd integer
2n + 1, we should have

n = kij, (3)

where kij is defined in (2). Furthermore, any couple (2i + 1, 2j + 1) of divisors of
2n + 1, with i ≤ j, must satisfy the following inequality

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On the structure of composite odd integers and prime numbers

2i + 1 ≤
√
2n + 1 ≤ 2j + 1 (equality holds iff i = j).

It follows that

1 ≤ i ≤ In =
1

2
(−1 +

√
2n + 1) ≤ j. (4)

Let In = {1, 2, ..., in}, with in = [In] =integer part of In.

The implications (1) state that the special form (2) of n is a necessary and
sufficient condition in order for any odd integer 2n + 1 to be a composite number.
Moreover, if 2n + 1 is a prime, then n cannot be put in the form (2). In the fol-
lowing theorem, given a generic n, necessary and sufficient conditions are given
for either n ∈ K or n ∈ N \ K.

Theorem 2.1. Let n ∈ N. Then

n ∈ K iff ∃i ∈ In : (2i + 1) | (n − i), (5)

n ∈ N \ K iff ∀i ∈ In : (2i + 1) ∤ (n − i). (6)

Proof. A non implicit relation between i and j is obtained by making explicit
the variable j in (3). In doing so, (3) is written in the form of a real homographic
function of the integer variable i

y = ϕn(i) =
n − i
2i + 1

, i ∈ In, y ∈ R. (7)

If for some i we have y ∈ N, then the pair (i, j) with j = y = [y] identifies a
couple of divisors of 2n + 1. Thus, in this way we can identify all the couples of
divisors of 2n + 1, if they exist. The theses follow directly from (7).

2

The in conditions assumed to have n ∈ N \ K are not superabundant. Conse-
quences of theorem 1 are the following two propositions.

If n ∈ K, then n has a number of representations n = kij just equal to the
number of couples of divisors of 2n + 1.

If n ∈ N \ K, then n can be expressed as

n = i + (2i + 1)qi + ri = kiqi + ri,

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G. Buffoni

with qi, ri ∈ N and 1 ≤ ri < 2i + 1, ∀i = 1, 2, ..., in.

Remark 2.1. (r1) ϕn(i) is decreasing with i from ϕn(1) = (n − 1)/3 to ϕn(in).

(r2) Equation y = ϕn(i) is a simple algorithm to compute the couples of divi-
sors of an integer 2n + 1, if they exist. It can also be used as a primality test. The
order of the number of operations is

√
n/2.

(r3) By direct calculation it follows that

2n + 1 = 2kij + 1 = (i + j + 1)
2 − (j − i)2.

This identity shows the well known fact that any composite odd integer may be
written as a difference of two squares. Possibly in more than two ways, while for
a prime only holds the decomposition 2n + 1 = (n + 1)2 − n2.

(r4) Let mq be the sequence of integers defined recursively by

mq = 2mq−1 + 1, q = 1, 2, ..., m0 = 0,

which implies that mq = 2q − 1. When q is a prime, mq is a Mersenne number.
Let q = 2r. Since m2r = 22r − 1 = (2r + 1)(2r − 1) is composite, we have that
m2r−1 ∈ K, r = 2, 3, ....

3 Geometrical implcations of the properties of n
The relation n = kij can be rewritten as

n = i(j + 1) + (i + 1)j, (8)

which can be interpreted as the determinant of the matrix

A =

(
i −j

i+1 j+1

)
.

Since det(A) = n, n is the area of the parallelogram with the column vectors

u = (i, j + 1)T , v = (−j, j + 1)T ,

as two of its sides. This property holds ∀(i, j) such that kij = n. In addition to
the equality of determinants, no other relationships have been recognized between
the matrices A related to the pairs (i, j) with kij = n.

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On the structure of composite odd integers and prime numbers

Moreover (8) shows that n is also the area of the union of the two rectangles
of sides i, (j + 1) and (i + 1), j.

Imagine that you are drawing, on a squared sheet, rectangles composed by an
odd integer number of squares. The side of the squares is assumed as linear unit
of measure, so that the rectangles have integer sides and area. Obviously, only
integers 2n + 1 with n satisfying (8) can represent rectangles. The number 2n + 1
is the area of the rectangle of sides (2i + 1), (2j + 1). Moreover, with the help of
(8), we can show that this rectangle is the union of five rectangles: two with sides
i, j + 1, two with sides i + 1, j, plus one unit square (Fig. 1).

Figure 1: Decomposition of the rectangle of sides 2i + 1, 2j + 1. i = 3, j = 5,
n = 3 · (5 + 1) + (3 + 1) · 5 = 38, 2n + 1 = 2 · 38 + 1 = 77, (2i + 1) · (2j + 1) =
7 · 11 = 77

Thus we have shown that, when n ∈ K, then n represents also the area of the
union of two rectangles, and 2n + 1 both the area of one rectangle and the union
of five rectangles.

When n /∈ K, then n can be represented as n = kiqi + ri, with ri ≥ 1, which
leads to the following representation for 2n + 1

2n + 1 = 2kiqi + 1 + 2ri = (2i + 1)(2qi + 1) + 2ri, 1 ≤ ri < 2i + 1.

Since n /∈ K, this expression can not be reduced to the product of two odd integers.

Let us now consider the problem of the localization of composite odd integers
2ν + 1, with 4 ≤ a ≤ ν ≤ n. The discrete relation n = kij is viewed in the
continuous form as n = κ(x, y) = x + y + 2xy, (x, y) ∈ R2, and in explicit
form it is written as

y = ϕn(x) =
n − x
2x + 1

, 1 ≤ x ≤ In. (9)

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G. Buffoni

ϕn(x) is decreasing with x from ϕn(1) = (n − 1)/3 to ϕn(In) = In.
Let the set Ω(a, n) be defined as

Ω(a, n) = {(x, y) ∈ R2 | 1 ≤ x ≤ In, (x ≤ y) ∨ (a ≤ κ(x, y) ≤ n)}. (10)

Figure 2: Set Ω(a, n): n = 100, a = 50. Continuous lines: y = ϕn(x) and
y = ϕa(x); dotted lines: x = 1 and y = x; asterisk: points (Ia, 0), (Ia, Ia); circle:
points (In, 0), (In, In). Different scales for x and y.

It is a bounded, closed and convex set in a plane. From its implicit definition
(10), it follows that Ω(a, n) (Fig. 2) can be defined explicity by the following
inequalities as the union of two sets

1 ≤ x ≤ Ia, ϕa(x) ≤ y ≤ ϕn(x), (11)

Ia < x ≤ In, x ≤ y ≤ ϕn(x). (12)
Both ϕa(x) and ϕn(x) are decreasing with x, and also their difference

∆n,a(x) = ϕn(x) − ϕa(x) =
n − a
2x + 1

, 1 ≤ x ≤ Ia. (13)

Thus, when n − a is small enough, we have that ∆n,a(x) < 1, and Ω(a, n) may
contain at the most one integer point. Other details on this argument can be found
in appendix.

4 Concluding remarks
It is worthy to draw a concluding remark. In other words, Theorem 2.1 states

that any prime has just one parent: if a generic n satisfies (6) in Theorem 2.1, then
it is the parent of 2n + 1. Otherwise it is a composite number.

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On the structure of composite odd integers and prime numbers

References
[1] G. Buffoni. On odd integers and their couples of divisors. Ratio Mathemat-

ica, 40: 87-111, 2021.

Appendix
Cosider the limit case a = 4 (Fig. 3, left), so that Ia = 1, ϕ4(1) = 1. Thus,

the first set (11) consists of only one point x = Ia = 1. Since ϕa(Ia) = Ia, (11)
and (12) reduce to

1 ≤ x ≤ In, x ≤ y ≤ ϕn(x).

Consider the case a large, i.e. n−a small (Fig. 3, right). Assume n−a << a.
The difference In − Ia may be written as

In − Ia = 0.5
√
2n + 1 (1 −

√
2a + 1

2n + 1
) = 0.5

√
2n + 1 (1 −

√
1 −

2(n − a)
2n + 1

).

From the assumption n − a << a we have 2(n − a)/(2n + 1) << 1, so that

In − Ia ≃
n − a

2
√
2n + 1

< 1. (14)

Thus, the second set (12) reduces to a very small interval.

Figure 3: Set Ω(a, n): n = 100. Left: limit case a = 4; right: case a = 90, so that
In − Ia < 1. Notes as in caption of Fig. 2

Consider now the set of points with integer coordinates. The set Ω∗(a, n) of
the pairs (i, j), such that 1 ≤ i ≤ j, a ≤ kij ≤ n, defined by

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G. Buffoni

Ω∗(a, n) = {(i, j) ∈ N | i = 1, 2, ..., In, (i ≤ j) ∨ (a ≤ kij ≤ n)}, (15)

is the set of the points (x, y) ∈ Ω(a, n) with x and y integer coordinates. Af-
terwards, it is straightforward to formulate the explicit definition of the pairs
(i, j) ∈ Ω∗(a, n). Then, Ω∗(a, n) is again defined as the union of two sets

i = 1, 2, ..., [Ia], j = J(i), J(i) + 1, ..., [ϕn(i)], (16)

where either J(i) = [ϕa(i)] + 1 when ϕa(i) /∈ N or J(i) = ϕa(i) when ϕa(i) ∈ N,
and

i = [Ia] + 1, ..., [In], j = i, i + 1, ..., [ϕn(i)]. (17)

When b−a is small enough, it follows that In−Ia < 1, so that either [In] = [Ia]
or [In] = [Ia] + 1. The second set (17) is missing when [In] = [Ia], and consists
of only one point when [In] = [Ia] + 1; thus (16) and (17) reduce to

i = 1, 2, ..., [In], j = J(i), J(i) + 1, ..., [ϕn(i)]. (18)

If ν ∈ {a, a + 1, ..., n} \ {kij | (i, j) ∈ Ω∗(a, n)}, then 2ν + 1 is a prime.

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