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Infinitely many composites

Published online by Cambridge University Press:  15 February 2024

Nick Lord
Tonbridge School, Kent TN9 1JP e-mail:
Des MacHale
School of Mathematics, Applied Mathematics and Statistics University College Cork, Cork, Ireland e-mail:


In number theory, we frequently ask if there are infinitely many prime numbers of a certain type. For example, if n is a natural number:

  1. (i) Are there infinitely many (Mersenne) primes of the form 2n − 1?

  2. (ii) Are there infinitely many primes of the form n2 + 1?

These problems are often very difficult and many remain unsolved to this day, despite the efforts of many great mathematicians. However, we can sometimes comfort ourselves by asking if there are infinitely many composite numbers of a certain type. These questions are often (but not always) easier to answer. For example, echoing (i) above, we can ask if there are infinitely many composites of the form 2p − 1 with p a prime number but (to the best of our knowledge) this remains an unsolved problem. Of course, it must be the case that there are either infinitely many primes or infinitely many composites of the form 2p − 1 and it seems strange that we currently cannot decide on either of them.

© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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