How to determine when the following sum will be prime?

Question

I was playing around with dates in my head and thought of the following prime number problem.

Problem:

The following (numerical) days of the month are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.

From this, we know the following (numerical) months are prime: Feb (2), Mar (3), May (5), July (7), Nov (11).

Interestingly, some upcoming prime years are 2027 and 2029.

Let $y$ be the year (four figures), $m$ the month, and $d$ the day. When is the next day that they are all prime and sum to a prime?

Question:

I think I have already shown that the sum must be odd by assuming that $y, m$ and $d$ are odd (which I now realise doesn't account for 2). However, I'm trying to prove that a sum of three primes can be prime and can't think of a way to go about it. Any ideas?

Answer 1

Using brute force:

Using $2027$ as year of course we can't have $2029$ as sum of day and month ($1+1$ are not primes)

The next prime is $2039$.

$2027 + 12 = 2039 \,$ so day + month must be $12$

$5 + 7 = 12$ and they are primes.

So the next date is the 7th of May 2027.