Computational number theory

1. Computational number theory is a branch of mathematics that focuses on using computer algorithms and techniques to study and solve problems related to numbers. It involves the utilization of computational tools to analyze number theoretic concepts and phenomena, such as prime numbers, factorization, modular arithmetic, and cryptographic schemes. Through the use of computational methods, researchers and mathematicians can explore complex number theoretic questions, develop efficient algorithms for solving mathematical problems, and analyze the behavior of various number sequences and properties. Computational number theory plays a crucial role in modern cryptography, data encryption, and the security of digital communication systems, making it a fundamental area of study in both mathematics and computer science.

Which algorithm is commonly used to find the greatest common divisor (GCD) of two integers?

A) Euclidean algorithm
B) Sieve of Eratosthenes
C) Fermat's Little Theorem
D) Binary Search

2. What is the Chinese Remainder Theorem used for in computational number theory?

A) Calculating factorials
B) Finding prime numbers
C) Converting decimals to fractions
D) Solving systems of simultaneous congruences

3. What is the smallest prime number?

A) 3
B) 5
C) 1
D) 2

4. What does the function Euler's Totient function count?

A) Number of positive integers less than n that are coprime to n
B) Number of divisors of n
C) Count of even numbers less than n
D) Number of prime factors of n

5. What is Wilson's Theorem?

A) The product of any k consecutive numbers is divisible by k!
B) Every number is a factorial of another number
C) The sum of consecutive odd numbers is always even
D) p is a prime number if and only if (p-1)! ≡ -1 (mod p)

6. How many prime numbers are there between 1 and 20 (inclusive)?

A) 9
B) 6
C) 8
D) 7

7. Which theorem states that every even integer greater than 2 can be expressed as the sum of two prime numbers?

A) Pythagorean Theorem
B) Goldbach's Conjecture
C) Fermat's Last Theorem
D) P vs NP Problem

8. What is a Sophie Germain prime?

A) Prime p such that 2p + 1 is also prime
B) Prime whose square root is prime
C) Prime number greater than 100
D) Prime with only 1 factor

9. What is a Mersenne prime?

A) Prime with exactly 2 factors
B) Perfect square that is prime
C) Prime number that is one less than a power of 2
D) Prime number greater than 1000

10. How is the Mobius function defined for a positive integer n?

A) μ(n) = n² - n for any positive integer n
B) μ(n) = 1 if n is even and 0 if n is odd
C) μ(n) = 1 if n is a square-free positive integer with an even number of distinct prime factors, μ(n) = -1 if n is square-free with an odd number of prime factors, and μ(n) = 0 if n has a squared prime factor
D) μ(n) = -1 if n is prime and 0 otherwise

11. What is the order of 2 modulo 11?

A) 10
B) 9
C) 5
D) 11

12. What is the term for a number that has no positive divisors other than 1 and itself?

A) Even number
B) Composite number
C) Odd number
D) Prime number

13. What is the divisor function σ(n) used to calculate?

A) Sum of all positive divisors of n
B) Number of perfect numbers less than n
C) Number of prime factors of n
D) Euler's Totient function value of n

14. Which concept in number theory involves finding integer solutions to linear equations in multiple variables?

A) Euler's theorem
B) Diophantine equations
C) Perfect numbers
D) Pell's equation

15. What does the value of the Legendre symbol (a/p) indicate, where p is an odd prime?

A) Number of divisors of p+a
B) Value of the function f(a, p) = a^p
C) Number of solutions to the equation a² = p (mod m)
D) Indicates whether a is a quadratic residue modulo p

16. What is a Niven number?

A) Integer that is divisible by the sum of its digits
B) Perfect number with prime factors
C) Even number less than 10
D) Prime number greater than 100

17. What is the common use of the Miller-Rabin primality test?

A) Finding the GCD of two numbers
B) Sorting numbers in descending order
C) Calculating the Fibonacci sequence
D) Checking primality of large numbers

18. What is the order of the group of integers modulo 7 under multiplication modulo 7?

A) 6
B) 5
C) 4
D) 7

19. What is the value of φ(12), where φ is Euler's totient function?

A) 6
B) 8
C) 10
D) 4

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