Twin Prime Factorial

Let p n p_{n} denote the n th n^\text{th} prime number and let Q n = p 1 × p 2 × p 3 × × p n Q_{n} = p_{1} \times p_{2} \times p_{3} \times \cdots \times p_{n} . For example, Q 1 = 2 , Q 2 = 2 × 3 , Q 3 = 2 × 3 × 5 Q_{1} = 2, Q_{2} = 2 \times 3, Q_{3} = 2 \times 3 \times 5 . Is the following true or false?

For any positive integer n n , at least one of Q n + 1 Q_{n} + 1 or Q n 1 Q_{n} - 1 is prime.

True False

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1 solution

Sam Bealing
May 1, 2016

If the statement for true then we would have a way of infinitely generating primes which clearly we don't so the answer is:

F a l s e \boxed{False}

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