Problems & Puzzles: Conjectures

Conjecture 83.  Two conjectures about primes

Dmitry Kamenetsky wrote on March 10, 2019

I have two conjectures:

C1) For any prime p>3 there exists a prime q<p, such that 2p+q is prime.

 

C2) For any prime p>3 there exists a prime q<p, such that 4p+q is prime.

 

I have checked that they work for p<10000, but I don't know if they always work.

Q. Can anyone prove C1/C2 or find a counter example?

On Dec 6, 2019, Dmitry wrote:

I think these conjectures are somehow related to the Goldbach conjecture and are thus difficult to prove. I have created sequences for them, which you may want to add:

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