Prime-o-phile Numbers!

Find all integers which can be represented as the sum of two primes and difference of two primes. (e.g.- 8=5+3=19-11)

#NumberTheory

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

All even numbers $\geq 4$ ; and all odd numbers $x$ such that $x-2$ and $x+2$ both are primes.
(If repetition of primes is allowed)

Yatin Khanna - 4 years, 5 months ago

Can you please explain how you arrived at the answer.

Aaron Jerry Ninan - 4 years, 5 months ago

To be honest; I played a big gamble there.
Whether every positive even integer can be written as sum and difference of two primes is actually an open problem (till my knowledge goes).
While, the second part can be easily seen. As the sum and difference are odd then there must be one odd and one even prime; and since 2 is the only even prime; the result follows.

Yatin Khanna - 4 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$