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User:FUNG Cheok Yin/A property of the set "2, 4, 6, 10, 12" (Proof of a statement by Ely Golden in Dec 2016)
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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 (which we would like to label as (I)):
- 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 ,
The Inequalities
Considering the part A and B, holds only if :
- for . (*)
It remains to find for which values of the inequality
holds.
Let
- , defined on
for all
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.