On a k-tuple conjecture for some explicit
integer sequences
Benoit Cloitre
September 2, 2012
Abstract
We study odd integer sequences satisfying an ⇠ n log n (n ! 1) like
the prime numbers and make conjectures related to the values of the
differences an+1 an . This led us to make conjectures for these sequences
similar to the twin prime conjecture or the k-tuple conjecture. Despite we
have many informations about theses sequences and convincing arguments
it turns out that proving these conjecture seems not easy.
Intoduction
The question of whether there exist infinitely many twin primes is still an open
question. This is the content of the twin prime conjecture, which states there are
infinitely many primes p such that p + 2 is also prime [Apo]. In 1849 de Polignac
made the more general conjecture that for every natural number k, there are infinitely many prime pairs (p, q) of consecutive primes such that p−q = 2k [Pol].
Mathematicians introduced many sequences behaving like primes using sieve
methods (Lucky numbers, Hawkins random primes, ...) and made similar conjectures than the twin prime conjecture, the Goldbach conjecture etc. Cramer
model also helped to suggest conjectures related to primes. But it appears few
explicit integer sequences behaving roughly like the primes were studied. In this
paper we consider sequences of odd integers given by simple explicit formulas
using the floor function and claim that they satisfy the analogue of the k-tuple
conjecture. In a first section we introduce and study the family of sequences of
odd integers (an )n 1 given by the formula:
⌫
log n
+1
an = 2 (n + sin n)
2
where > 0 is any real value. We then observe that the differences an+1 an
equal 2 zero time, finitely many times or infinitely many times depending on .
Thus we make a Polignac conjecture (conjecture 1) and a more general k-tuple
conjecture (conjecture 2) for these sequences. We provide also some heuristic
1
arguments supporting the conjectures. In a second section we try to give a
quantitative version of our conjectures 1 and 2.
1
A family of parametrized sequences
Let
> 0 and define the sequence of odd integers (an )n
⌫
log n
an = 2 (n + sin n)
+1
2
1
as follows
This family of sequence is interesting since they behave roughly like the
primes, i.e. an ⇠ n log n (n ! 1) and the differences an+1 an appear to follow
roughly similar patterns than the difference of consecutive primes. However they
are not always strictly increasing since the difference can sometime be zero but
it doesn’t matter for our purpose. Hereafter we make the conjecture 1, which
is an analogue of the Polignac prime conjecture, and the conjecture 2, which is
an analogue of the more general k-tuple conjecture.
1.1
Conjecture 1 (the Polignac conjecture for an )
This is a Polignac conjecture for this family of sequences, i.e. depending on the
values of the differences an+1 an take infinitely many times the same even
value. Namely let m 0 and define
✓
◆ 1
1
c0 := 2 sin
= 1.04....
2
Then we claim
1. 0 <
2.
↵ (m) := |{k 2 N | ak+1
< c0 ) 8m
c0 ) 8m
ak = 2m}|
0, ↵ (m) < 1
1, ↵ (m) = 1
In fact only line 2. is a conjecture since one can prove the line 1. using heuristic
arguments described below. Let us state another conjecture, which is similar
to k-tuple conjecture for primes. But here there is obviously no congruence
obstruction and all possible constellations are considered.
1.2
Conjecture 2 (the k-tuple conjecture for an )
Suppose we have k positive integers satisfying 1  b1 < b2 < ... < bk then we
claim
1. 0 <
2.
< c0 ) |{n 2 N | 8i 2 {1, 2, ..., k} , an+i = an + 2bi }| < 1
c0 ) |{n 2 N | 8i 2 {1, 2, ..., k} , an+i = an + 2bi }| = 1
2
1.3
Heuristic arguments
Both conjectures seem true due to the following fact. Let
an = (n + sin n) log n + h(n)
where 0  h(n)  1. Then we have
✓
1
an+1 an = (n+1+ sin(n+1)) log(n) + + O(n
n
2
)
◆
(n+ sin n) log n+h(n+1) h(n)
hence
an+1
an = (1 + (sin(n + 1)
sin n)) log(n) + 1 + h(n + 1)
h(n) + O(n
1
)
Now the equation
1 + (sin(x + 1)
✓
◆
1
sin x) = 0 ) cos x +
=
2
has real positive solutions if
given by
⇢n = cos
1
2 sin 12
✓
1
1
2 sin
2
◆
1
2 sin 12
and in this case solutions are
1
!
+ 2⇡n
Now we can expect that for infinetely many n, ⇢n is sufficiently closed to an
integer such that letting N = b⇢n c we have
aN +1
aN = 1 + h(N + 1)
h(N ) + O(N
1
logN)
Furthermore we think h(n + 1) h(n) is equidistributed in [ 1, 1] hence we
could find infinitely many N such that
aN +1
aN = 0
and similarly infinitely many N such that
aN +1
aN = 2
And the twin pair conjecture would be true for the sequence an . Regarding the
Polignac conjecture 1 we return above and we can expect that for any fixed
m 1 there are infinetely many n with ⇢n sufficiently closed to an integer (but
not too much!) such that letting N = b⇢n c we have
(1 + (sin(N + 1)
sin N )) log(N ) = 2m + O(N
And thus we would have infinitely many N such that
aN +1
aN = 2m
And the Polignac conjecture for an would hold.
3
1
log N )
Remark
It is interesting to note that for
< c0 but closed to c0 there are a huge
number of twin pairs. For instance if = 1 we found twin pairs 1033 but no
more
1040 . This is a good example of the danger to infer conjectures from
experiments!
In an other hand, since an ⇠ n log n, a random integer of size n would be
a term of the sequence (an )n 1 with probability log1 n for any > 0 (at first
glance). So using classical heuristic arguments (allowing us to support the
twin prime conjecture), one would expect to have infinitely many twin pairs in
(an )n 1 but we know it isn’t the case for < c0 . So one should be cautious
using too simple probabilistic arguments.
2
Quantitative version of the conjectures 1 and 2
We fix
1
= c0 = 2 sin 12
= 1.04... and we define
f (N ) := |{k 2 N : ak  N, ak+1
ak = 2}|
From the conjecture 1 we suspect limN !1 f (N ) = 1. Here we try to obtain
estimations on the behaviour of f and we believe
f (N ) ⇠ C1
N
(log N )
2
(N ! 1)
where C1 ' 2.5. Hence me make the conjecture 3.
2.1
Conjecture 3
In general for any m
⇡2m (N ) :=
n
0 we claim there is a constant Cm > 0 such that we have
2
(p, q) 2 (an )n
1 , (p, q)  N | p
Experimental support
2
We plot ⇡2m (n) (lognn) for various m.
4
q = 2m
o
⇠ Cm
N
(n ! 1)
(log N )2
2
⇡2m (n) (lognn) for m = 0, 1, 2, 3 (red, black, blue, green respectively).
And regarding the conjecture 2 we also believe the following conjecture 4
holds.
2.2
Conjecture 4
Suppose we have k positive integers satisfying 1  b1 < b2 < ... < bk and let
n
o
k
⇡k (N ) := (p1 , ..., pk ) 2 (an )n 1 , (p1 , ..., pk )  N | pi+1 pi = 2bi
The we claim there is a constant Ck (b1 , b2 , ..., bk ) > 0 such that we have
⇡k (N ) ⇠ Ck (b1 , b2 , ..., bk )
N
(n ! 1)
(log N )k
Concluding remarks
We don’t provide precise conjectures related to the values of the constants Cm
or Ck (b1 , b2 , ..., bk ) and it would be nice to have them in closed form. In this
study we just point out that it is somewhat difficult to prove k-tuple conjectures
like despite the sequences are very well known with explicit formulas. To us the
twin prime conjecture or the k-tuple conjecture are much more harder than the
Goldbach conjecture (some people said these are conjectures of same difficulty).
Indeed the twin pair conjecture require oscillatory properties of the sequence
whereas Goldbach’s like conjectures are less sensitive to this aspect (we were able
to state a general conjecture on the subject [Clo] using density arguments and
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didn’t find obvious counter example). Here we need fine probabilistic arguments
in order to prove the sequence an contains infinitely many k-tuples and we
wonder whether arguments in section 1 can be made rigorous. We have also
investigated many other sequence like
⌫
log n
an = 2 (n + w(n))
+1
2
where w(n) = o(n) is explicitely defined and observe similar things.
Regarding the prime numbers and the k-tuple conjecture we need to have
p(n+1)
p(n)
w(n) ⇠ log(n+1)
log(n) to be as closed as we wish of zero in order to have
infinitely many k-tuples like described in the section 1. The result in [Gol] that
is
lim inf
n!1
p(n + 1)
(log p(n))
1/2
p(n)
log log p(n)
<1
is therefore promising. But to us a probabilistic approach seems necessary to
prove the k-tuple conjecture for primes and we are not totally confident about
the possibility of a proof.
References
[Apo] T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 6
[Pol] Alphonse de Polignac, Recherches nouvelles sur les nombres premiers,
Comptes Rendus des Séances de l’Académie des Sciences, 1849.
[Clo] B. Cloitre, A propos de la conjecture de Goldbach, on line draft, 2012.
[Gol] D. A. Goldston, J. Pintz and C. Y. Yıldırım, Primes in Tuples III: On the
difference p(n + n)−p(n), Funct. Approx. Comment. Math. XXXV (2006),
79–89
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w(n,m)=if(n<0,0,s=u=1;while(b(s)<n,s++;u=u+if(b(s+1)-b(s)-2*m,0,1));u)
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