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) Binary Search
C) Sieve of Eratosthenes
D) Fermat's Little Theorem

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

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

3. What is the smallest prime number?

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

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

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

5. What is Wilson's Theorem?

A) Every number is a factorial of another number
B) The product of any k consecutive numbers is divisible by k!
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) 7
C) 8
D) 6

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

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

8. What is a Sophie Germain prime?

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

9. What is a Mersenne prime?

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

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

A) μ(n) = -1 if n is prime and 0 otherwise
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) = n² - n for any positive integer n

11. What is the order of 2 modulo 11?

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

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

A) Prime number
B) Even number
C) Odd number
D) Composite 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) Pell's equation
B) Euler's theorem
C) Diophantine equations
D) Perfect numbers

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

16. What is a Niven number?

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

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

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

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

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

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

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

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