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 :

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:

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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

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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?")

***

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.
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
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^(1-c) 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 k-th 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 k-almost 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 ...

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