Even Integers not Expressible as Sum of 3, 5 or 7 with Prime

Theorem

The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins:

$98, 122, 124, 126, 128, 148, 150, \ldots$

These are the primes which coincide with the upper end of a prime gap greater than $6$.

These can be found at:

$89$ to $97$: prime gap of $8$

$113$ to $127$: prime gap of $14$

$139$ to $149$: prime gap of $10$

and so on.

We have that:

$\ds 98$

$=$

$\ds 19 + 79$

$\ds 122$

$=$

$\ds 13 + 109$

$\ds 124$

$=$

$\ds 11 + 113$

$\ds 126$

$=$

$\ds 13 + 113$

$\ds 128$

$=$

$\ds 19 + 109$

and so on.

$\blacksquare$