[seqfan] Re: On the infinity of Sophie Germain primes

[Benoît Jubin]

On Fri, Aug 14, 2015 at 5:22 PM, Benoît Jubin <benoit.jubin at gmail.com>
wrote:

> I do not see how this follows from Motohashi's (not Hiratoshi) result
> http://matwbn.icm.edu.pl/ksiazki/aa/aa17/aa1736.pdf
> However, a weaker result that follows from Motohashi's and is enough to
> produce prime partitionable numbers with the same recipe as in the
> Trotter-Erdos article is the following, if I'm not mistaken:
> there exist infinitely many prime pairs p_1, p_2 with p_2 = 1 (mod p_1)
> and p_1 < p_2 < p_1^2 -p_1.
>

To be a bit more precise:
Following Motohashi, for k>0, denote by P(k) the least prime congruent to 1
modulo k.
Then by Motohashi's result, there exist infinitely many primes q such that
P(q) < C q^{1.7} for some constant C.
In particular, there exist infinitely many primes q such that P(q) < q^2-q
(for q large enough, one has C q^{1.7} < q^2-q).
Note that for these primes, q < P(q).

For such q, write, (p_1,p_2) = (q,P(q)).
We prove that d = p_1 + p_2 is prime partitionable.
Indeed, let n_1 = d p_1 p_2 and n_2 = d times all primes less than d and
different from p_1 and p_2.
Let d_1, d_2 > 0 be such that d_1 + d_2 = d and suppose that gcd(n_2,d_2)=1.
Since d_2 < d, the prime factors of d_2 can only be p_1 or p_2.
Since d = p_1 + p_2 < p_1^2, then d_2 is equal to 1 or p_1 or p_2.
Therefore d_1 = p_1 + p_2 - d_2 is divisible by p_1 or p_2 or p_1
respectively, so it is not coprime to n_1.

Regards,
Benoît


>
> Best regards,
> Benoît
>
> On Fri, Aug 14, 2015 at 3:37 PM, Chris <cgribble263 at btinternet.com> wrote:
>
>> Dear Seqfans,
>>
>>
>>
>> https://en.wikipedia.org/wiki/Sophie_Germain_prime and
>> http://mathworld.wolfram.com/SophieGermainPrime.html both refer to
>> Hoffman,
>> P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search
>> for Mathematical Truth.
>> <http://www.amazon.com/exec/obidos/ASIN/0786863625/ref=nosim/weisstein-20
>> >
>> New York: Hyperion, p. 190, 1998 in which it is stated that the conjecture
>> that there are infinitely many Sophie Germain primes remains unproven.
>>
>>
>>
>> However, in W. T. Trotter, Jr. and Paul Erdős,
>> <https://www.renyi.hu/~p_erdos/1978-49.pdf> When the Cartesian product of
>> directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142
>> DOI:10.1002/jgt.3190020206, p. 141, it is proved that there are infinitely
>> many prime-partitionable numbers.  The proof relies on a theorem by
>> Hirotashi from which follows that there exist infinitely many prime pairs
>> p1, p2 with p1 > 3 for which p2 = 2p1 + 1.  Doesn't this imply that there
>> are infinitely many Sophie Germain primes?
>>
>>
>>
>> Best regards,
>>
>> Chris Gribble
>>
>>
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>
>