There is no solution. For two numbers to add to an odd total, one of them must be even. If that number is also a prime it must be 2.
But if one of the numbers is 2 and the two sum to 289, the other must be 287. But 287 is not a prime and the question required two primes to be added.
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17 and 17
The prime number (factor) of 289 is: 17
289 = 17 x 17
289 does.
289 is an odd number so it must be the sum of an odd and an even number. The only even prime number is 2 so the other number must be 287, which is not prime as it is divisible by 7, so your question has no answer in this universe.