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Continued fractions constructed
from prime numbers
Marek Wolf
e-mail: mwolf@ift.uni.wroc.pl
Abstract
We give 50 digits values of the simple continued fractions whose denom-
inators are formed from a) prime numbers, b) twin primes, c) primes
of the form m2 + 1 and Mersenne primes. All these continued frac-
tions belong to the set of measure zero of exceptions to the Khinchin
Theorem.
1 Introduction
Let a0 be an integer and let ak, k = 1, 2, . . . , n are positive integers. Then
a = [a0; a1, a2, a3, . . . , an] = a0 +
1
a1 +
1
a2 +
1
a3 + . . .
1
an
(1)
is the simple (i.e. with all nominators equal to 1) finite continued fraction. Let
pn
qn
= [a0; a1, a2, a3, . . . , an] (2)

be the n-th convergent of a. If the sequence pn/qn converges to some limit a when
n →∞ then we say that the infinite continued fraction
a = [a0; a1, a2, a3, . . .] (3)
converges to the same limit a. The sufficient and necessary condition for convergence
of the continued fraction (3) is the divergence of the series:
∞
∑
n=1
an (4)
see e.g. [4, Theorem 10, p.10]. If the infinite continued fraction is convergent
then the values of the convergents pk/qk approximate the value of a with accuracy
1/qkqk+1 [4, Theorem 9, p.9]:
∣
∣
∣
∣
a− pkqk
∣
∣
∣
∣
< 1qkqk+1
. (5)
Rational numbers have finite continued fractions, quadratic irrationals have periodic
infinite continued fractions and all remaining irrational numbers have non-periodic
continued fractions.
Khinchin has proved that [4, p.93].
lim
n→∞
(
a1a2 . . . an
)
1
n =
∞
∏
m=1
{
1 + 1m(m + 2)
}log2 m
≡ K0 ≈ 2.685452001 . . . (6)
is a constant for almost all real r, see also e.g.[2, §1.8]. The exceptions are rational
numbers, quadratic irrationals and some irrational numbers too, like for example the
Euler constant e = 2.7182818285 . . ., but this set of exceptions is of the Lebesgue
measure zero. The constant K0 is called the Khinchin constant. All presented
below continued fractions belong to this exceptional set of irrationals for which the
geometric means of the denominators ak:
K(n) =
(
a1a2 . . . an
)
1
n (7)
will tend to infinity.
2

Because there is infinity of primes as well as (conjectured) infinity of some families
of primes, we can construct non-periodic infinite continued fractions taking as the
denominators an just those primes. We consider here the following cases: the set of
all primes 2, 3, 5, 7, . . ., twin primes, primes given by the quadratic form m2 +1 and
Mersenne primes.
2 The set of all primes
Let us put an = pn where pn denotes the n-th primes: [0; 2, 3, 5, 7, 11, 13, . . .]. As
there is an infinity of primes the condition (4) is fulfilled and let us denote the limit
of the continued fraction by
u = [0; 2, 3, 5, 7, 11, 13, . . .] =
1
2 +
1
3 +
1
5 +
1
7 +
1
11 + . . .
(8)
Using PARI system [7] and all 1229 primes up to 10000 it is possible to obtain over
8000 digits of the above continued fraction in just a few seconds because
[0; 2, 3, 5, 7, 11, 13, . . . , 9973] = 3.38592889 . . .× 10
4297
7.83177791 . . .× 104297 (9)
and the product of qkqk+1 on the rhs of (5) is larger than 108500. The first 50 digits
of u reads:
u = 0.43233208718590286890925379324199996370511089688 . . . . (10)
This number is not recognized at the Symbolic Inverse Calculator (http://pi.lacim.uqam.ca/eng/)
maintained by Simone Plouffe.
3

It is possible to obtain analytically the geometrical means of the denominators
in (8). It is well known (see e.g. [1, Chap.4]), that the Chebyshev function θ(x)
behaves like:
θ(x) =
∑
p≤x
log(p) = x +O(
√
x). (11)
Thus skipping the error term we have
n
∏
k=1
pk = epn. (12)
It is well known that [5, Sect. 2.II.A] that
pn = n log(n) + n(log log(n)− 1) + o
(n(log log(n)
n
)
. (13)
Again skipping the error term we can write for the geometrical means of the denom-
inators (7) the estimation:
(
a1a2 . . . an
)
1
n =
(
n
∏
k=1
pk
)
1
n
= (epn)
1
n ∼ n →∞ (14)
thus the continued fraction u belongs to the set of measure zero of exceptions to the
Khinchin Theorem.
3 Twin primes
The twin prime conjecture states that there are infinitely many pairs of primes
(tn, tn+1) differing by two: tn+1 − tn = 2. Let pi2(x) denote the number of pairs
of twin primes (tn, tn+1) smaller than x. Then the conjecture B of Hardy and
Littlewood [3] on the number of prime pairs p, p+ d applied to the case d = 2 gives,
that
pi2(x) ∼ C2
∫ x
2
u
log2(u)du = C2
x
log2(x) + . . . , (15)
where C2 is called “twin constant” and is defined by the following infinite product:
C2 ≡ 2
∏
p>2
(
1− 1(p− 1)2
)
= 1.32032363169 . . . (16)
4

If there is indeed (as everybody believes) an infinity of twins, then the continued
fraction (we count here 5 two times as it is a customary way of defining the Brun’s
constant [6])
u2 = [0, 3, 5, 5, 7, 11, 13, 17, 19, . . .] (17)
should be infinite and convergent. Again performing calculations in PARI and using
primes < 10000 we found here 205 twin pairs (but only 409 different primes) and
first 50 digits of the continued fraction (17) are
u2 = 0.31323308098694591263078648647217280043925117451 . . . (18)
Because there is much less terms in u2 for primes< 10000 than in u the value of
u2 was obtained with accuracy about 2900 digits. We have checked using Plouffe’s
Symbolic Inverse Calculator (http://pi.lacim.uqam.ca/eng/), that this constant is
not recognized as a combination of other mathematical quantities.
Because twin primes are sparser than all primes we have tn > pn thus in view
of (14) the geometrical means (3 · 5 . . . tn)1/n will diverge even faster, hence the
continued fraction u2 belongs to the set of exceptions to the Khinchin Theorem.
4 Primes of the form m2 + 1
Now let us consider the set of prime numbers
Q = {2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, . . .} (19)
given by the quadratic polynomial m2 + 1 and let qn denote the n-th prime of this
form. By the conjecture E of Hardy and Littlewood [3] the number piq(x) of primes
q < x of the form q = m2 + 1 is given by
piq(x) ∼ Cq
√x
log(x) , (20)
where
Cq =
∏
p≥3
(
1− (−1)
(p−1)/2
p− 1
)
= 1.372813462818246009112192696727 . . . (21)
5

As this conjecture remains unproved there is no doubt in its validity. Thus let us
create the presumably infinite continued fraction by identifying an = qn, n ≥ 1:
uq = [0; 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, . . .] (22)
Using 841 primes of the form m2+1 smaller than 108 and performing the calculations
in PARI we get over 11000 digits of uq as the ratio on the rhs of (5) was < 10−11700.
First 50 digits of uq reads:
uq = 0.45502569980199468718020210263808421898137687948 . . . (23)
There is no known formula analogous to (11) for primes of the form m2 +1, but
because qn ≥ pn the geometrical means of 2 ·5 ·17 . . . qn will diverge faster than (14).
It is possible to obtain very rough speed of divergence of (2 ·5 ·17 . . . qn)1/n. Namely,
inverting piq(qn) = n and making use of (20) we get:
qn ∼
(2n log(n/Cq)
Cq
)2
+ 2 log
( n
Cq
)
log log
( n
Cq
)
(24)
It follows that 2 · 5 · 17 . . . qn grows faster than 2n(n!)2/C2nq and the Stirling formula
gives that (2 · 5 · 17 . . . qn)1/n will grow faster than n2 and again uq is the exception
to the Khinchin theorem.
5 Mersenne primes
The Mersenne primes Mn are the primes of the form 2p − 1 where p is a prime,
see e.g. [5, Sect. 2.VII]. Only 47 primes of this form are currently known. Again
there is no proof of the infinitude of Mn but there is a common belief that as there
are presumably infinitely many even perfect numbers thus there is also an infinity
of Mersenne primes. Let us define the supposedly infinite and convergent continued
fraction uM by taking an = Mn:
uM = [0; 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, . . .] (25)
6

We have calculated the value of uM using the first 14 Mersenne primes 3, 7, 31, . . . ,
2607 − 1 and we get over 700 digits of uM; first 50 digits of uM are:
uM = 0.31824815840584486942596202748140694243806236564 . . . (26)
Of course in view of exponential grow of Mn the continued fraction uM is again the
exception to the Khinchin Theorem.
It is possible to consider other families of primes, like Sophie Germain primes
(it is conjectured that there are infinitely many of them), regular primes of which it
was conjectured that e−1/2 ≈ 61% of all prime numbers are regular etc.
References
[1] W. Ellison and F. Ellison. Prime Numbers. John Wiley and Son, 1985.
[2] S. Finch. Mathematical Constants. Cambridge University Press, 2003.
[3] G. H. Hardy and J. E. Littlewood. Some problems of ‘Partitio Numerorum’ III:
On the expression of a number as a sum of primes. Acta Mathematica, 44:1–70,
1922.
[4] A. Y. Khinchin. Continued Fractions. Dover Publications, New York, 1997.
[5] P. Ribenboim. The Little Book of Bigger Primes. Springer-Verlag New York,
Inc, 2004.
[6] D. Shanks and J. Wrench, John W. Brun’s constant. Mathematics of Computa-
tion, 28(125):293–299, 1974.
[7] The PARI Group, Bordeaux. PARI/GP, version 2.3.2, 2008. available from
http://pari.math.u-bordeaux.fr/.
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