Prime Problem

Number Theory Level 3

Take some time to read the statements below.

\([1]\). It is impossible for \(p\), \(p+2\) and \(p+4\) to be all prime numbers where \(p\) is a prime number greater than \(3\).

\([2]\). It is possible for both \(8p-1\) and \(8p+1\) to be prime when \(p\) is a prime number.

\([3]\). If \(p\) is a prime greater than \(3\), then \(p^2-1\) is always divisible by \(12\)

Which of these statements are true?

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.


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