Exercise · ZW-INFINITUDE-OF-PRIMES-009 · ★★☆ Intermediate
Using the list 2, 3, 5, 7, 11, what is the constructed number and a prime not in the list?
Choose one option.
Show answer
- D. 2311 and 2311
Explanation
The constructed number is \(2311\), which is prime and not in the list, so it serves as the new prime itself. The option giving \(2310\) confuses the product with the constructed number.
Review the concept: Infinitude of primes
The infinitude of primes is Euclid's theorem stating that the set of prime numbers is infinite: given any finite list of primes \(p_1,\dots,p_n\), there is a prime not on it.
\[p\mid p_1p_2\cdots p_n+1\ \Longrightarrow\ p\neq p_i\ \text{for all } i\]
More practice problems: infinitude of primes
- ZW-INFINITUDE-OF-PRIMES-001 Which statement summarizes Euclid's theorem about primes?
- ZW-INFINITUDE-OF-PRIMES-002 State the idea of Euclid's proof by contradiction.
- ZW-INFINITUDE-OF-PRIMES-003 What number is constructed in Euclid's proof from the primes 2, 3 and 5?
- ZW-INFINITUDE-OF-PRIMES-004 According to Euclid's argument, what property does N = p_1 p_2 cdots p_n + 1 have with respect to t…
- ZW-INFINITUDE-OF-PRIMES-005 Which statement about the list 2, 3, 5, 7 is correct?
- ZW-INFINITUDE-OF-PRIMES-006 Compute N = 2· 3· 5· 7· 11 + 1 and determine its smallest prime divisor.
- ZW-INFINITUDE-OF-PRIMES-007 What is the remainder of 2· 3· 5· 7· 11· 13 + 1 when divided by 7?
- ZW-INFINITUDE-OF-PRIMES-008 If p_1,dots,p_n are primes and N = p_1cdots p_n + 1, which statement about N is true?