All prime numbers except 2 are odd.

Case 1: One of the prime numbers is 2:

If one of the prime numbers is 2, the other must be 9, which is not a prime number.

Case 2: Neither of the prime numbers is 2:

Then both prime numbers are odd. Let x and y be odd numbers. Then x = 2k + 1 for any integer k (ignoring the restriction for primeness) and y = 2l + 1 for any integer l.

Then x - y = 2k + 1 - (2l + 1) = 2k - 2l = 2(k - 1). Since k - l is an integer, 2(k - l) is an even integer, and thus not 7.

Therefore the are no two prime numbers such that p1 - p2 = 7. QED.

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