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Post Closed as "Not suitable for this site" by Steven Landsburg, Andreas Blass, Michael Albanese, Shahrooz, Dag Oskar Madsen
This is basically rolling back to the initial version because the 2nd version edit significantly changes the scope of the question and, in particular, to a large extent invalidates the current accepted answer.
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edit approved Aug 7, 2022 at 21:42
John Omielan
edit approved Aug 7, 2022 at 21:42
John Omielan
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Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Let 𝑝 be a positive integer and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

added 11 characters in body
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Kelvin
Kelvin
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Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Let 𝑝 be a positive integer and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Source Link
Kelvin
Kelvin
  • 15
  • 2

Digit sum of a prime number

Let 𝑝 be a positive integer and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)