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

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

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

3. What is the smallest prime number?

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

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

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

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

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

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

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

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

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

11. 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

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) Number of perfect numbers less than n
D) Sum of all positive divisors of n

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

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

14. What is a Niven number?

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

15. 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 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
C) μ(n) = 1 if n is even and 0 if n is odd
D) μ(n) = -1 if n is prime and 0 otherwise

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

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

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

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

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

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

19. What is the order of 2 modulo 11?

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

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