Exercise · ZW-PRIME-NUMBER-010 · ★★★ Advanced
For how many integers \(n>0\) is \(n^2-1\) prime?
Choose one option.
Show answer
- B. Exactly one
Explanation
\(n^2-1=(n-1)(n+1)\). For \(n=1\) it is 0; for \(n\ge3\) both factors exceed 1, so it is composite. Only \(n=2\) gives 3, which is prime.
Review the concept: Prime number
A prime number \(p\) is an integer greater than 1 whose only positive divisors are 1 and itself.
\[p>1,\quad d>0,\ d\mid p\ \Rightarrow\ d\in\{1,p\}\]
More practice problems: prime number
- ZW-PRIME-NUMBER-001 Which of these numbers is prime?
- ZW-PRIME-NUMBER-002 How many prime numbers are less than 30?
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- ZW-PRIME-NUMBER-004 What is the prime factorization of 360?
- ZW-PRIME-NUMBER-005 To decide whether 97 is prime by trial division, what is the largest prime you must test?
- ZW-PRIME-NUMBER-006 Which of these statements about primes is true?
- ZW-PRIME-NUMBER-007 How many positive divisors does 60 have?
- ZW-PRIME-NUMBER-008 What is the smallest prime greater than 100?