Quiz Bowl You got to know it: Statements about prime numbers (Mathematics)
1. This theorem states that every positive integer greater than 1 has a unique prime factorization, which is the product of prime numbers in a specific order, although the order of factors does not matter. For example, the prime factorization of 80 is 2⁴·5. For 15 points, name this theorem.
Answer: Fundamental Theorem of Arithmetic
2. This theorem, often attributed to Euclid, proves that there are infinitely many prime numbers. The proof uses the fact that the product of any finite list of primes, plus one, must either be prime itself or divisible by primes not in the original list. For 15 points, name this theorem.
Answer: Euclid’s Theorem
3. This theorem states that if p is a prime number and a is any integer, then ap≡amod pa^p \equiv a \mod pap≡amodp, meaning that ap−aa^p - aap−a is divisible by p. It is widely used in probabilistic primality testing. For 15 points, name this theorem.
Answer: Fermat's Little Theorem
4. This theorem states that a number p is prime if and only if (p−1)!≡−1mod p(p - 1)! \equiv -1 \mod p(p−1)!≡−1modp. It was first used by Ibn Al-Haytham and later named after John Wilson. For 15 points, name this theorem.
Answer: Wilson’s Theorem
5. This theorem, named after Euclid and Euler, states that every even perfect number can be written as 2p−1(2p−1)2^{p - 1}(2^p - 1)2p−1(2p−1), where p is prime and 2p−12^p - 12p−1 is a Mersenne prime. For 15 points, name this theorem.
Answer: Euclid–Euler Theorem
6. This approximation describes the behavior of the prime counting function π(n)\pi(n)π(n), which gives the number of primes less than or equal to n. According to the prime number theorem, π(n)\pi(n)π(n) is approximately n/lognn / \log nn/logn as n becomes large. For 15 points, name this theorem.
Answer: Prime Number Theorem
7. This conjecture, proposed by Christian Goldbach, claims that every even integer greater than 2 can be expressed as the sum of two prime numbers. For 15 points, name this conjecture.
Answer: Goldbach's Conjecture (Strong Goldbach Conjecture)
8. This unproven hypothesis about the Riemann zeta function states that all of its nontrivial zeroes lie on the critical line in the complex plane, where the real part of each zero is 1/21/21/2. For 15 points, name this hypothesis.
Answer: Riemann Hypothesis
9. This conjecture posits that there are infinitely many pairs of prime numbers that differ by exactly 2, such as 3 and 5 or 101 and 103. For 15 points, name this conjecture.
Answer: Twin Prime Conjecture
10. This sequence of numbers is composed of primes that can be written in the form 2p+12p + 12p+1 where p is prime. These primes were studied by Sophie Germain long before the advent of public key cryptography. For 15 points, name these primes.
Answer: Safe Primes