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Computational number theory
Contributed by: Wyatt
  • 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) Binary Search
B) Euclidean algorithm
C) Fermat's Little Theorem
D) Sieve of Eratosthenes
  • 2. What is the Chinese Remainder Theorem used for in computational number theory?
A) Solving systems of simultaneous congruences
B) Finding prime numbers
C) Converting decimals to fractions
D) Calculating factorials
  • 3. What is the smallest prime number?
A) 5
B) 3
C) 1
D) 2
  • 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) Number of divisors of n
D) Count of even numbers less than 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) Pythagorean Theorem
D) Goldbach's Conjecture
  • 8. What is a Sophie Germain prime?
A) Prime with only 1 factor
B) Prime whose square root is prime
C) Prime number greater than 100
D) Prime p such that 2p + 1 is also prime
  • 9. What is a Mersenne prime?
A) Prime with exactly 2 factors
B) Prime number greater than 1000
C) Prime number that is one less than a power of 2
D) Perfect square that is prime
  • 10. 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) = 1 if n is even and 0 if n is odd
D) μ(n) = n2 - n for any positive integer n
  • 11. What is the order of 2 modulo 11?
A) 10
B) 5
C) 9
D) 11
  • 12. What is the term for a number that has no positive divisors other than 1 and itself?
A) Composite number
B) Odd number
C) Even number
D) Prime number
  • 13. What is the divisor function σ(n) used to calculate?
A) Number of prime factors of n
B) Number of perfect numbers less than n
C) Sum of all positive divisors 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) Perfect numbers
C) Euler's theorem
D) Diophantine equations
  • 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) Indicates whether a is a quadratic residue modulo p
C) Value of the function f(a, p) = ap
D) Number of solutions to the equation a2 = p (mod m)
  • 16. What is a Niven number?
A) Perfect number with prime factors
B) Integer that is divisible by the sum of its digits
C) Prime number greater than 100
D) Even number less than 10
  • 17. What is the common use of the Miller-Rabin primality test?
A) Checking primality of large numbers
B) Calculating the Fibonacci sequence
C) Sorting numbers in descending order
D) Finding the GCD of two numbers
  • 18. What is the order of the group of integers modulo 7 under multiplication modulo 7?
A) 4
B) 5
C) 7
D) 6
  • 19. What is the value of φ(12), where φ is Euler's totient function?
A) 8
B) 10
C) 4
D) 6
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