Problems & Puzzles:
Conjectures
Conjecture
49.
n = pq +rs
Patrick Capelle sends the following
conjectures:
Conjecture A : Every natural number n > 1 can be written as n = p.q
+ r.s , where p, q, r, s are primes or 1.
Conjecture B : Every natural number n > 33 is the sum of two
semiprimes
Questions :
1. Do you have some comments about the two
conjectures ?
2. Can you prove the conjecture A or
find a counterexample ?
3. Can you prove the conjecture B or
find a counterexample ?
Contributions came from Javier Falcó
Benavent & Farideh Firoozbakht:
***
Javier wrote:
Conjecture A : Every natural number
n > 1 can be written as n = p.q +
r.s , where p, q, r, s are primes or 1.
If n = 2 = 1*1 + 1*1
If n = 2k for all k>1, we can choose q = 1 and s = 1. We obtain
n = 2k = p + r
It was conjectured by Goldbach.
If n = 3 = 2*1 + 1*1
If n = 5 = 3*1 + 2*1
If n = 2k + 1 for all k>2, we choose q = 2 and s = 1. We obtain
n = 2p + r
It was conjectured by Levy (1963) and called Levi´s Conjecture. We can
find it in:
http://mathworld.wolfram.com/LevysConjecture.html
***
Farideh wrote:
Conjecture B is interesting
but about four years ago Lior Manor has asked it (please
see the comment line of A072966: "Is this sequence
finite?")
***
Later (13/10/06) Capelle wrote (ready?):
Question 1.
Is this sequence finite ?
Lior Manor was asking the good question.
It is important to mention it. Not yet an assertion, but the
first fruits of
a promising tree.
I would like to thank Farideh Firoozbakht for this information
about this
beautiful conjecture B.
I asked recently (Oct 8) on the forum of discussion Primenumbers
if there
exist other references.
(
http://tech.groups.yahoo.com/group/primenumbers/message/18383
). I did not
receive an answer.
Hence, the Manor's contribution should be regarded as the main
starting
point of this important subject.
Why not to give the name of Manor to the conjecture B ?
The conjecture B is a particular case of the conjecture A for n
> 33.
Other conjectures/theorems already known are also connected to
particular
cases of the conjecture A :
1. p = 1, q > 1, r = 1, s > 1 : all the even numbers > 2 are the
sum of
two primes (Goldbach’s conjecture, in the Euler’s version).
This conjecture is equivalent to the proposal "every even number
4n, with n
> 1, is the sum of two even semiprimes" [*].
Moreover, the Goldbach's conjecture implies that every even
number > 6 is
the sum of two semiprimes [**].
In the conjecture B, all the even numbers n > 33 are concerned !
2. p = 1, q > 1, r > 1 : every sufficiently large even number is
the sum of
either two primes, or a prime and a semiprime (Chen’s theorem).
3. p = 1, q > 1, r = 1, s > 1, n >= N very high : almost all
even numbers
can be written as the sum of two primes (Vinogradov ; Estermann
; Chen and
Wang).
4. p = 1, q > 1, r = 1, s > 1, n < N very high : "most" even
numbers are
the sum of two primes (Hugh Montgomery and Robert Charles
Vaughan). They
showed that there exist positive constants c, C such that for
all
sufficiently large numbers N, every even number less than N is
the sum of
two primes, with at most CN^(1c) exceptions. In particular, the
set of even
integers which are not the sum of two primes has density zero.
5. p = 1, q > 2, r = 1, s > 2 : all the even numbers > 4 are the
sum of two
odd primes (variant of the Goldbach's conjecture).
6. p = 1, q > 2, r = 1, s > 2, q <> s : all the even numbers > 6
are the
sum of two distinct primes (“Extended Goldbach’s conjecture” of
Joseph L.
Pe, equivalent to the Sebastian Martin Ruiz’s conjecture “for
all n >= 4
there exists an integer k , with 1 <= k <= n 1, such that phi(n^2
– k^2) =
(n 1)^2 – k^2 where Phi is the Euler totient function“).
7. p = 1, q > 2, r = 1, s > 2, q <> s, n = 2.t with t prime > 3
: all the
even semiprimes > 6 are the sum of two distinct primes
(adaptation from
the Zumkeller’s conjecture “the number of ways to represent the
kth prime
as arithmetic mean of two other primes is > 0 when k > 2”)
[***].
8. p = 2, r = 3, q <> s : all the prime numbers > 3 are of the
form 2q +
3s, where q and s are primes or 1 (Papadimitriou’s conjecture).
9. p = 2, q > 2, r = 3, s > 2, q <> s : all the prime numbers >=
19 are of
the form 2q + 3s, where q and s are odd primes (Firoozbakht’s
conjecture,
which is a particular case of the conjecture B for the prime
numbers > 33)
[****].
10. p = 2, q > 1, r = 1 , s > 1 : all the odd numbers >= 7 are
of the form
2q + s, where q and s are primes (Levy's conjecture, which is a
stronger
version of the weak Goldbach's conjecture) [*****].
If we define an 'Extended Semiprime' as a semiprime whose each
factor is
equal to a prime number or 1 (i.e., an Extended Semiprime is a
semiprime, a
prime number or 1), we obtain a simple formulation for the
conjecture A :
Every natural number n > 1 can be written as n = e + f , where e
and f are
Extended Semiprimes.
Examples :
1. The even numbers > 2 are the sum of two primes (Goldbach),
but ALL the
even numbers are the sum of two Extended Semiprimes.
2. Every Extended Semiprime > 1 is the sum of two Extended
Semiprimes.
[*] 2n = q + s, with n > 1 (Goldbach's conjecture).
==> 2.2n = 2.(q+s)
==> 4n = 2q + 2s = sum of two even semiprimes for n > 1.
Reverse implication :
4n, with n > 1, is the sum of two even semiprimes.
A semiprime is even iff at least one of the two prime factors is
equal to 2.
==> There exists two primes, q and s, such that 4n = 2q + 2s,
with n > 1.
==> 2.2n = 2.(q + s), with n > 1.
==> 2n = q + s, with n > 1.
[**] 2n = q + s, with n > 1 (Goldbach's conjecture).
==> t.2n = t.(q + s), with t prime.
==> t.2.n = tq + ts.
==> 2.m = tq + ts, with m = t.n
m covers all the natural numbers > 3 when t prime and n > 1
(think to the
sieve of Eratosthenes).
==> Every even number 2.m, with m > 3, is the sum of two
semiprimes.
Or, if you prefer :
2n = q + s, with n > 1
==> t.2n = t.(q + s), with t prime.
==> t.2.n = tq + ts = sum of two semiprimes for each t prime.
==> 4n = 2q +2s, 6n = 3q +3s, 10n = 5q + 5s, 14n = 7q + 7s, ...
The set of the even numbers 4n, 6n, 10n, 14n, 22n, ...
represents all the
even numbers > 6.
In conclusion, if the 'strong' Goldbach's conjecture is true,
then every
even number > 6 is the sum of two semiprimes.
[***] Note that by the conjecture B, all the semiprimes > 33 are
the sum of
two semiprimes.
It implies that every even semiprime > 33 is at the same time
the sum of two
distinct primes and the sum of two semiprimes.
In other words, the even semiprimes > 33 can be written as the
sum of two
'Extended Semiprimes' in two different ways ...
[****] The Firoozbakht’s conjecture leads to some interesting
generalizations :
1. All the prime numbers greater or equal to 19 are of the form
2q + 3s,
where q is prime and s is odd prime.
2. All the odd natural numbers > 33 are the sum of an even
semiprime and an
odd semiprime (p = 2, q > 1, r > 2, s > 2).
In the conjecture B, all the odd numbers n > 33 are concerned !
3. Every sufficiently large kalmost prime, with k > 0, is the
sum of two
(k+1)almost primes.
[*****] More precisely s > 2, because s is never even. The odd
numbers >= 7
are in fact the sum of an odd prime plus twice a prime.
If we modify the formulation of the Levy's conjecture in the
sense that s >
2 and if the Firoozbakht's conjecture is generalized in the
sense that q is
prime and s is odd prime (see above), then we obtain that the
prime numbers
>= 7 are of the form 2q + s in the same time that the prime
numbers >= 19
are of the form 2q + 3s, with q prime and s odd prime in both
cases. It
means that all the prime numbers >= 19 are simultaneously of two
distinct
forms : the form 2q + s and the form 2q + 3s, where q is prime
and s is odd
prime.
Let t be a prime number >= 19, and t = 2q + s = 2q' + 3s'.
Comparison
between s and 3s' : they are odd but s is prime and 3s' is
composite, which
implies that they are different. If s and 3s' are different,
then the even
numbers 2q and 2q' are different, which implies that q and q'
are different.
Finally, the four terms 2q, 2q', s and 3s' are different. We
could say that
all the prime numbers t > = 19 can be written as a sum of two
'Extended
Semiprimes' in two different ways ...
***
