A Late Night Conversation

Two years ago, we were up talking at about One.

She: Why aren't you replying for last two minutes?


Me: My computer just stopped responding.


She: Isn't that just an excuse to not talk to me?


Me: No, dear. I just calculated that <LargePrimeX> + <LargePrimeY> = <YourPhoneNumber> where <LargePrimeX> and <LargePrimeY> are prime numbers.


She: You do all these up so late?


Me: Yup...


She: Had it been any other girl, she'd have gone mad. Only because it's me, I've been in sound mental health for so long!


I recently realised that the last four digits of her phone number can be expressed as a sum of two primes in 103 ways.

What is the maximum possible value of the last four digits of her phone number?


Details and Assumptions:

  • PrimeA + PrimeB and PrimeB + PrimeA are counted as the same solution.

  • The primes need not be distinct.

  • 22 can be written as a sum of two primes in three ways.

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