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Let \(p\) be a prime number and let \(n_1\) and \(n_2\) be two integers such that:$$n_1 = p + 1\,000 \quad \text{and} \quad n_2 = p + 2\,000$$
  1. By reasoning modulo 3, show that the only possible value for \(p\) such that \(n_1\) and \(n_2\) are both prime numbers is 3.
  2. Can \(n_1\) and \(n_2\) actually both be prime?

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