2 and 3 are the only pair.
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Nobody invented them. Some prime numbers were found to be next-but-one to one another, and these were called twin primes.
There are no two primes whose sum, difference, product or quotient is 97. However, there may be some non-standard binary operation such that two primes can be combined to make 97.
Some pairs of prime numbers are: 3, 5 11, 17 7, 23
There are no two consecutive primes whose sum, difference, product or quotient is 56.However, there may be some non-standard binary operation such that two primes can be combined to make 56.