An elementary construction is given of an infinite sequence of natural numbers, having at least two different decompositions as sum of primes and no prime number appears in more than one of them

Goldoni, Luca

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A NOTE ON AN ADDITIVE PROPERTY OF PRIMESLUCA GOLDONIAbstract. An elementary construction is given of an infinite se-quence of natural numbers, having at least two different decom-positions as sum of primes and no prime number appears in morethan one of them.1. IntroductionPrime numbers are the “building blocks” for the multiplicative struc-ture of the set of all natural numbers. They arise in a very natural waywhen one deals with factorization of integers and one of the most im-portant theorem about them is the so called “Fundamental theorem ofArithmetic”(FTA) witch states that each natural number n > 1 canbe written in an essentially unique way as a product of distinct primesfactors. But what is the behavior of primes in the “ additive context”?It is easy to check that a theorem like FTA can not be true. In factall the integers of the form p + 5, being p a prime, can be written asp + 2 + 3 as well. In this way one provides an infinite family of in-tegers admitting at least two different decompositions. But in all thenumbers is present a “fixed” summand: the number 5. What if fixedsummands are not allowed? Does there exist an infinite family of inte-gers admitting multiple “essentially different” decomposition? Here by“essentially different” we mean that the primes used in the decomposi-tions of one of these integers are not used in the decompositions of theothers ones. A positive answer to the question can be given with theuse of standard tools of Analytic Number Theory. Using Chebishevbounds of pi(x) it is possible to prove the existence of such a family ofintegers. The aim of this note is just to give an elementary constructionof such a sequence, requiring only very basic notions of Calculus. Tothis purpose the proof is based on a combinatorial result by Erdos andBenkoski [1], that in spite its depth, admits a nice proof using nothingmore than High School algebra.Date: February 6, 2009.1991 Mathematics Subject Classification. 11A41,11P99.Key words and phrases. prime additive decomposition, elementary proof.Dipartimento di matematica. Universita` di Trento. Dottorato di ricerca XXIIIciclo.12 LUCA GOLDONI2. The main resultTheorem 1. There exist an infinite set A = {a1, a2, · · ·an · · ·} ⊆ Nsuch that for every an ∈ A there exist two different sets An,Bn ofprimes such thatAn ∩ Bn = ∅ andan =∑p∈Anpan =∑p∈BnpMoreover, if Dn = An ∪ Bn then Dn ∩Dm = ∅ ∀n 6= mThe last claim tell us that all the primes used for an are smaller thanthe primes used for an+1 so the representations are really different.For the sake of completion it will be given even an proof based onstandard techniques of Analytic Number Theory witch ultimately canbe regarded as a corollary of a work of Chebyshev [2]. He provednamely the following theorem:Theorem 2 (Chebyshev bounds). If pi(x) denotes the number of primesless or equal to a given real number x, then there are two positives con-stants, a and A such thataxlog x6 pi (x) 6Axlog xfor all x large enough.On the basis of this theorem an analytical proof runs as follow:Proof. Let x large enough,then if we consider the interval [x/100, x/2]the number of primes in such interval ispi(x/2)− pi(x/100)≫xlog xas follows from Chebyshev’s bounds. If P denotes the set of all primesandA = {(p, q) : p, q ∈ P, p < q, p, q ∈ [x/100, x/2]}we have|A| ≫x2log2 xBut for each (p, q) ∈ A it is p + q ≤ x so ,for x large enough, not allsums can be distinct, therefore there must exist two different pairs ofprimes having the same sum. Varying x, we get infinitely many suchexamples. A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 3As one can see, it is possible achieve very quickly the main resultalong this way, but it require the knowledge of the Chebyshev’s boundsand even though they are not too difficult to prove, they requires a quitelong preparatory work.3. The pillar of the elementary proofIn 1974 P.Erdo¨s and S. Benkoski found the following result:Lemma 1 (Erdo¨s-Benkoski). Let A a finite set of positive integers suchthat for all B,C ∈ P(A)B 6= C ⇒∑x∈Bx 6=∑y∈Cythen∑a∈A1a< 2where P(A) denotes the set of all subsets of a given set.Loosely speaking one can see a ”competition ” between the numberand the size of elements of the set A and the fact that all the possi-ble subsets must have different sums. This theorem can be proved,invarious ways. One of them, as one can find in [1], uses High SchoolAlgebra only, and so it can be regarded as fully elementary.4. The elementary proof of Theorem 1Proof. One starts with the a well known result related with the seriesof the reciprocals of primes. This series as Euler proved is a divergentseries. That means∑p∈P1p= +∞so we can find a first prime p1 such that(1)∑p∈P2≤p≤p11p≥ 2LetS1 = {2, 3 · · ·p1}If we consider all the subsets of S1 we notice that it’s not possiblefor them to have all different sums because if so, by Lemma (1)∑p∈P2≤p≤p11p< 24 LUCA GOLDONIagainst (1). So there are two different subsets of S1, say A1 and B1such that ∑A1 =∑B1Leta1 =∑A1 =∑B1Now, we choose a prime q1 such that q1 > a1 and then the first primep2 such that ∑p∈Pq1≤p≤p21p≥ 2With the same argument as before, we find two different subsets A2,B2 ofS2 = {q1 · · · p2}such that ∑A2 =∑B2So we can choosea2 =∑A2 =∑B2and so on.If one define A = {a1, a2 · · ·} one has that A is a set with the propertyof Theorem 1 and by construction it is plain that the sets Dn = An∪Bnare pairwise disjoint. If an is expressed in two different ways as sum ofprimes, obviously, all the common summands can be deleted, obtaininga new a′n witch has two different representations without any prime incommon. In other words it is always possible think that An e Bn aredisjoint sets.Note 1. We can formulate Theorem 1 using odd primes only.In thiscase for every n we have that the numbers of elements of An and of Bnhave the same parity.Figure 1 illustrates the situation. Herean ={pr1 + pr2 + pr3qr1 + qr2 + qr3andan+1 ={ps1 + ps2 + ps3qs1 + qs2 + qs3Of course the draw is not in scale, but only suggests the meaning ofthe main result.A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 5Figure 1References[1] Benkoski, S. J.; Erdo¨s, P. On weird and pseudoperfect numbers. Math. Comp.28 (1974), 617–623.[2] Chebyshev, P. L. ”Me´moir sur les nombres premiers.” J. math. pures appl. 17,1852[3] Honsberger R. Mathematical Gems III, Chapter 17 ( Dolciani MathematicalExpositions, No.9)Universita` di Trento, Dipartimento di Matematica, v. Sommarive14, 56100 Trento, ItalyE-mail address : goldoni@science.unitn.it

A note on an additive property of primes.

University of Trento : UNITN-eprints

A NOTE ON AN ADDITIVE PROPERTY OF PRIMES
LUCA GOLDONI
Abstract. An elementary construction is given of an infinite se-
quence of natural numbers, having at least two different decom-
positions as sum of primes and no prime number appears in more
than one of them.
1. Introduction
Prime numbers are the “building blocks” for the multiplicative struc-
ture of the set of all natural numbers. They arise in a very natural way
when one deals with factorization of integers and one of the most im-
portant theorem about them is the so called “Fundamental theorem of
Arithmetic”(FTA) witch states that each natural number n > 1 can
be written in an essentially unique way as a product of distinct primes
factors. But what is the behavior of primes in the “ additive context”?
It is easy to check that a theorem like FTA can not be true. In fact
all the integers of the form p + 5, being p a prime, can be written as
p + 2 + 3 as well. In this way one provides an infinite family of in-
tegers admitting at least two different decompositions. But in all the
numbers is present a “fixed” summand: the number 5. What if fixed
summands are not allowed? Does there exist an infinite family of inte-
gers admitting multiple “essentially different” decomposition? Here by
“essentially different” we mean that the primes used in the decomposi-
tions of one of these integers are not used in the decompositions of the
others ones. A positive answer to the question can be given with the
use of standard tools of Analytic Number Theory. Using Chebishev
bounds of pi(x) it is possible to prove the existence of such a family of
integers. The aim of this note is just to give an elementary construction
of such a sequence, requiring only very basic notions of Calculus. To
this purpose the proof is based on a combinatorial result by Erdos and
Benkoski [1], that in spite its depth, admits a nice proof using nothing
more than High School algebra.
Date: February 6, 2009.
1991 Mathematics Subject Classification. 11A41,11P99.
Key words and phrases. prime additive decomposition, elementary proof.
Dipartimento di matematica. Universita` di Trento. Dottorato di ricerca XXIII
ciclo.
1
2 LUCA GOLDONI
2. The main result
Theorem 1. There exist an infinite set A = {a1, a2, · · ·an · · ·} ⊆ N
such that for every an ∈ A there exist two different sets An,Bn of
primes such thatAn ∩ Bn = ∅ and
an =
∑
p∈An
p
an =
∑
p∈Bn
p
Moreover, if Dn = An ∪ Bn then Dn ∩Dm = ∅ ∀n 6= m
The last claim tell us that all the primes used for an are smaller than
the primes used for an+1 so the representations are really different.
For the sake of completion it will be given even an proof based on
standard techniques of Analytic Number Theory witch ultimately can
be regarded as a corollary of a work of Chebyshev [2]. He proved
namely the following theorem:
Theorem 2 (Chebyshev bounds). If pi(x) denotes the number of primes
less or equal to a given real number x, then there are two positives con-
stants, a and A such that
ax
log x
6 pi (x) 6
Ax
log x
for all x large enough.
On the basis of this theorem an analytical proof runs as follow:
Proof. Let x large enough,then if we consider the interval [x/100, x/2]
the number of primes in such interval is
pi(x/2)− pi(x/100)≫
x
log x
as follows from Chebyshev’s bounds. If P denotes the set of all primes
and
A = {(p, q) : p, q ∈ P, p < q, p, q ∈ [x/100, x/2]}
we have
|A| ≫
x2
log2 x
But for each (p, q) ∈ A it is p + q ≤ x so ,for x large enough, not all
sums can be distinct, therefore there must exist two different pairs of
primes having the same sum. Varying x, we get infinitely many such
examples. 
A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 3
As one can see, it is possible achieve very quickly the main result
along this way, but it require the knowledge of the Chebyshev’s bounds
and even though they are not too difficult to prove, they requires a quite
long preparatory work.
3. The pillar of the elementary proof
In 1974 P.Erdo¨s and S. Benkoski found the following result:
Lemma 1 (Erdo¨s-Benkoski). Let A a finite set of positive integers such
that for all B,C ∈ P(A)
B 6= C ⇒
∑
x∈B
x 6=
∑
y∈C
y
then
∑
a∈A
1
a
< 2
where P(A) denotes the set of all subsets of a given set.
Loosely speaking one can see a ”competition ” between the number
and the size of elements of the set A and the fact that all the possi-
ble subsets must have different sums. This theorem can be proved,in
various ways. One of them, as one can find in [1], uses High School
Algebra only, and so it can be regarded as fully elementary.
4. The elementary proof of Theorem 1
Proof. One starts with the a well known result related with the series
of the reciprocals of primes. This series as Euler proved is a divergent
series. That means
∑
p∈P
1
p
= +∞
so we can find a first prime p1 such that
(1)
∑
p∈P
2≤p≤p1
1
p
≥ 2
Let
S1 = {2, 3 · · ·p1}
If we consider all the subsets of S1 we notice that it’s not possible
for them to have all different sums because if so, by Lemma (1)
∑
p∈P
2≤p≤p1
1
p
< 2
4 LUCA GOLDONI
against (1). So there are two different subsets of S1, say A1 and B1
such that ∑
A1 =
∑
B1
Let
a1 =
∑
A1 =
∑
B1
Now, we choose a prime q1 such that q1 > a1 and then the first prime
p2 such that ∑
p∈P
q1≤p≤p2
1
p
≥ 2
With the same argument as before, we find two different subsets A2
,B2 of
S2 = {q1 · · · p2}
such that ∑
A2 =
∑
B2
So we can choose
a2 =
∑
A2 =
∑
B2
and so on.
If one define A = {a1, a2 · · ·} one has that A is a set with the property
of Theorem 1 and by construction it is plain that the sets Dn = An∪Bn
are pairwise disjoint. If an is expressed in two different ways as sum of
primes, obviously, all the common summands can be deleted, obtaining
a new a′n witch has two different representations without any prime in
common. In other words it is always possible think that An e Bn are
disjoint sets.

Note 1. We can formulate Theorem 1 using odd primes only.In this
case for every n we have that the numbers of elements of An and of Bn
have the same parity.
Figure 1 illustrates the situation. Here
an =
{
pr1 + pr2 + pr3
qr1 + qr2 + qr3
and
an+1 =
{
ps1 + ps2 + ps3
qs1 + qs2 + qs3
Of course the draw is not in scale, but only suggests the meaning of
the main result.
A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 5
Figure 1
References
[1] Benkoski, S. J.; Erdo¨s, P. On weird and pseudoperfect numbers. Math. Comp.
28 (1974), 617–623.
[2] Chebyshev, P. L. ”Me´moir sur les nombres premiers.” J. math. pures appl. 17,
1852
[3] Honsberger R. Mathematical Gems III, Chapter 17 ( Dolciani Mathematical
Expositions, No.9)
Universita` di Trento, Dipartimento di Matematica, v. Sommarive
14, 56100 Trento, Italy
E-mail address : goldoni@science.unitn.it


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