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

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

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

3. What is the smallest prime number?

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

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

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

5. What is Wilson's Theorem?

A) The product of any k consecutive numbers is divisible by k!
B) The sum of consecutive odd numbers is always even
C) Every number is a factorial of another number
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) 6
B) 9
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) Fermat's Last Theorem
B) P vs NP Problem
C) Goldbach's Conjecture
D) Pythagorean Theorem

8. What is a Sophie Germain prime?

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

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

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

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

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

11. What is a Mersenne prime?

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

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

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

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

14. What is a Niven number?

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

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

A) 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
B) μ(n) = -1 if n is prime and 0 otherwise
C) μ(n) = n² - n for any positive integer n
D) μ(n) = 1 if n is even and 0 if n is odd

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

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

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

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

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

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

19. What is the order of 2 modulo 11?

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

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