Proving Sum of Two Primes is Never Twice a Prime

[sachinism]

Prove that sum of two primes can never be twice a prime

p.s: find the actual edited q in 4th post belowsorry for the mistake

 

[Outlined]

'twice a prime' ?

 

[sjb-2812]

Prove that sum of two primes can never be twice a prime

Counter examples: 2 (a prime) + 2 (a prime) = 4 = twice 2 (a prime); or if you object to using the same prime twice or more: 7 + 19 = 26 = 2 x 13

 

[sachinism]

ah my bad

this is the correct q

Show that sum of two consecutive primes is never twice a prime

 

[sjb-2812]

What is the difference between consecutive primes?

 

[ramsey2879]

It seems to me that this question is redundant of an earlier thread on November 12which depended upon the fact that the average of two consecutive odd primes can not be a prime.

 

[sachinism]

@ramsey

can you give me the link of that thread please

 

[epkid08]

Let $$(j, k)\in\mathb{N}^2$$. Without loss of generality, assume $$j < k$$.


From here, it's safe to assume that: $$\forall (j, k)$$, $$j < \frac{j + k}{2} < k$$.


Then, $$p_n < \frac{p_n + p_{n+1}}{2} < p_{n+1}$$.


Since $$p_n$$ and $$p_{n+1}$$ are consecutive primes, $$\frac{p_n + p_{n+1}}{2}$$ cannot be prime.

 

[ramsey2879]

@ramsey

can you give me the link of that thread please

https://www.physicsforums.com/showthread.php?t=447162