A property of the set "2, 4, 6, 10, 12" (Proof of a statement by Ely Golden in Dec 2016)
link to A048597
On Dec 05 2016, Ely Golden stated in http://oeis.org/A048597 :
- It appears that 2, 4, 6, 10, 12 are all the numbers
with the property that every number
in the range
that is coprime to
is also prime.
We would like to state it as a lemma:
- Lemma I:
- 2, 4, 6, 10, 12 are all the numbers
with the property that every number
in the range
that is coprime to
is also prime.
Motivation
2, 4, 6, 10 and 12 are both small integers; it makes us naturally curious whether there are some inequalities relating to the number of primes −
− and the number of integers relatively prime −
− to compose a proof on Golden's statement.
Outline of the Proof
If
is odd and
,
is relatively prime to
but composite. Hence the possible
's satisfy the stated property are all even.
Lemma I implies the following inequality:
-
![{\displaystyle \varphi (2n)-\varphi (n)\leq \pi (2n)-\pi (n)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/db3d69ddd49d76a23aab7ff55c856fffef9fb880)
- does not hold for all even
.
The following three results in analytic number theory will be used:
-
for
.
-
for
.
-
,for even
and
, where the numbers
in the product are running from all odd prime divisors of
.
The first two results came from the paper Approximate formulas for some functions of prime numbers (J. Barkley Rosser ,Lowell Schoenfeld, Illinois J. Math., p.64-94 (1962), online view). The third result comes naturally from the definition of
.
Part A
For
, where:
is odd , larger than 1; and
a positive integer; we have
-
-
-
Part B
For
,
-
-
![{\displaystyle ={\frac {2n}{\ln 2+\ln n}}-1.25506\cdot {\frac {n}{\ln n}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d48c922104711eb925525429a586486abbc42e99)
-
-
The Inequalities
Considering the part A and B,
holds only if :
-
for
. (*)
It remains to find for which values of
the inequality
-
![{\displaystyle {\frac {1}{2}}\leq {\frac {2}{0.69315+\ln n}}-{\frac {1.25506}{\ln n}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ab76e344167feec414fbec48f823eb4fcbe998ae)
holds.
Let
-
, defined on ![{\displaystyle n\in [3,\infty ]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fd76794798bc3d747608094018744e0543845ed6)
for all
-
![{\displaystyle K(n)={\frac {2\ln n-0.69315-\ln n}{(0.69315+\ln n)(\ln n)}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/31278db3602294b7e2479591e539e843ce3e5841)
-
![{\displaystyle ={\frac {\ln n-0.69315}{(0.69315+\ln n)(\ln n)}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cdc35010cc47bfc7297ecdb06777358260de0f47)
Being evaluated numerically, the maximum value of
is roughly 0.24753, at
.
Hence
for all
.
To sum up,
for
.
Hence
may hold only for
.
To Conclude
Reviewing the constraints in the previous parts, all we need to do now, is checking for the small interval
= 2 to 17,
even, which
's have the stated property in Lemma I.
- 2: 2 < 3 < 4; 3 is prime.
- 4: 4 < 5 < 7 < 8; 5 and 7 are prime.
- 6: 6 < 7 < 9 < 11 < 12; 7 and 11 are prime, 9 and 6 have a common divisor 3.
- 8: 8 < 9 < ... < 16; 9 and 8 are relatively prime but 9 is composite.
- 10: 10 < 11 < 13 < 15 < 17 < 19 < 20; 11, 13, 17 and 19 are prime, 15 and 10 have a common divisor 5.
- 12: 12 < 13 < 15 < 17 < 19 < 21 < 23 < 24; 13, 17, 19 and 23 are prime, 12, 15 and 21 have a common divisor 3.
- 14: 14 < 15 ... < 28; 15 and 14 are relatively prime but 15 is composite.
- 16: 16 < 17 < 19 < 21 < ... < 25 < 27 < ... < 32; as a power of 2, 16 is relatively prime to 21, 25 and 27 but the three are composite.
After the above enumerations, we know those
's satisfying the property suggested by Golden, what in the beginning of this disclosure, are 2, 4, 6, 10 and 12.
Draft began on Aug 11 2017
Finished on Dec 24 2020