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Stellar Prime Numbers
Heitor Baldo
To cite this version:
Heitor Baldo. Stellar Prime Numbers. 3 pages. 2014.
HAL Id: hal-00984742
https://hal.archives-ouvertes.fr/hal-00984742v1
Submitted on 28 Apr 2014 (v1), last revised 28 Jan 2015 (v3)
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STELLAR PRIME NUMBERS
HEITOR BALDO
Abstract. In this paper we introduce a new class of prime numbers:
the stellar prime numbers. These prime numbers have been defined as
the primes that are sums of a odd number of distinct odd primes. These
sets of primes have many interesting properties.
1. Introduction
There are many type of primes in mathematics, each one defined in a particular
way. For example, the Sophie Germain primes are the primes p such that 2p + 1 is
n
prime, the Fermat primes are the primes 22 + 1, the twin primes, that (p, p + 2)
are both primes, and etc. Now we will analyze a new kind of prime numbers, called
stellar primes.
Definition 1.1. A α-stellar prime number is a prime number that is the sum of
α distinct odd primes, when α is a odd number such that α ≥ 3.
The case α = 1 is trivial because all odd prime is a sum of one prime number:
himself.
Notation 1.2. Let Gα denote the set of the α-stellar primes and P denote the
set of all primes.
Then some sets of the α-stellar primes are:
G1 = P − {2} = {3, 5, 7, 11, 13, ...}
G3 = {19, 23, 29, 31, 37, ...}
G5 = {43, 47, 53, 59, 61, ...}
G7 = {79, 89, 97, ...}
G9 = {127, 139, ...}
G11 = {199, ...}
G13 = {283, ...}
and so on.
Note that the smallest prime that is sum of seven distinct odd primes is 79. The
smallest that is sum of nine distinct odd primes is 127 and so on.
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Note also that 3+5+7+11+13+19+23+29+31+37=195, is the smallest number
that’s sum of 11 distinct odd prime numbers, and isn’t a prime. So the next number
prime after 195 is 199, and 3+5+7+11+13+19+23+29+31+41=199, hence 199 is
the smallest prime number that is sum of 11 distincts odd prime numbers. For 19
we have: 3+5+7=15, that isn’t prime, and as 9 isn’t prime too ⇒ the next prime to
consider after 15 can’t be 17, so is the 19 ⇒ 3+5+11=19.
The letter G, “G”, was used to represents the set of stellar primes to indicate
Galaxies: a collection of many stars. Sometimes we’ll call the set Gα of α-Galaxy.
2. The α-Stellar Primes
Now we expose some conjectures about the stellar prime numbers without, however, worry about their proofs.
Conjecture 2.1. There are infinite α-stellar numbers, for all α odd integer.
Conjecture 2.2. If pα represents the smallest α-stellar prime with α ≥ 7, then
it is give to the formula
pα = 2pα−2 − pα−4 + 12n
when n ∈ N.
Conjecture 2.3. For all odd integer n we have the infinite inclusion
P ⊃ G1 ⊃ G3 ⊃ G5 ⊃ . . . ⊃ Gn ⊃ . . . .
Conjecture 2.4. Every prime number p ≥ 19 is sum of 3 distincts odd primes,
i.e., every prime number p ≥ 19 is a 3-stellar prime.
The difference of the conjecture above and the Goldbach’s Weak Conjecture
(proved in [4]) is: if p is 3-stellar then p is the sum of 3 distincts odd primes.
Notation 2.5. Let NO denote the set NO = {n ∈ N : n ≥ 3 and n is odd}.
Notation 2.6. Let G denote the set G = {Gα : α ∈ NO }.
Proposition 2.7. The set G is enumerable.
Proof. Consider the application
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φ : α ∈ NO −→ Gα ∈ G
It’s clear this application is a bijection. Conjecture 2.8. Let G∞ be the set of ∞-stellar prime numbers that are sum
of infinite distinct odd primes. Then G∞ 6= ∅.
Check on the web the Galaxies of stellar prime numbers:
http://stellarprimes.atspace.co.uk [5]
Consideration: the name stellar prime number was choosen because we can write
them as illustrate the figure below.
Figure 1. Pictorial representation of the stellar prime numbers.
References
[1]
[2]
[3]
[4]
[5]
D. Zagier. Die ersten 50 Millionen Primzahlen. Basel: Birkhuser, 1977.
W. J. Ellison. Prime numbers. Paris: Hermann, 1985.
P. Ribenboim. The little books of big primes. New York, NY: Springer, 1991.
H. A. Helfgott. Major arcs for Goldbach’s Problem. arXiv:1305.2897v1, 2013.
Galaxies of Stellar Prime Numbers - http://stellarprimes.atspace.co.uk.
Date: April, 27th 2014.
Universidade Estadual de Campinas (UNICAMP), Instituto de Matemática, Estatı́stica e Computação Cientı́fica, IMECC, Brazil.
E-mail adress: [email protected] / [email protected]
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