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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) Fermat's Little Theorem
B) Euclidean algorithm
C) Sieve of Eratosthenes
D) Binary Search
  • 2. What is the Chinese Remainder Theorem used for in computational number theory?
A) Finding prime numbers
B) Converting decimals to fractions
C) Calculating factorials
D) Solving systems of simultaneous congruences
  • 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) Count of even numbers less than n
B) Number of divisors of n
C) Number of prime factors of n
D) Number of positive integers less than n that are coprime to n
  • 5. What is Wilson's Theorem?
A) p is a prime number if and only if (p-1)! ≡ -1 (mod p)
B) Every number is a factorial of another number
C) The sum of consecutive odd numbers is always even
D) The product of any k consecutive numbers is divisible by k!
  • 6. How many prime numbers are there between 1 and 20 (inclusive)?
A) 7
B) 6
C) 9
D) 8
  • 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 a Mersenne prime?
A) Prime number greater than 1000
B) Perfect square that is prime
C) Prime with exactly 2 factors
D) Prime number that is one less than a power of 2
  • 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 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
  • 11. What is the order of 2 modulo 11?
A) 10
B) 9
C) 11
D) 5
  • 12. 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
  • 13. What is the divisor function σ(n) used to calculate?
A) Number of perfect numbers less than n
B) Number of prime factors of n
C) Euler's Totient function value of n
D) Sum of all positive divisors 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) Indicates whether a is a quadratic residue modulo p
C) Number of solutions to the equation a2 = p (mod m)
D) Value of the function f(a, p) = ap
  • 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) Calculating the Fibonacci sequence
C) Sorting numbers in descending order
D) Checking primality of large numbers
  • 18. What is the order of the group of integers modulo 7 under multiplication modulo 7?
A) 5
B) 7
C) 6
D) 4
  • 19. What is the value of φ(12), where φ is Euler's totient function?
A) 4
B) 6
C) 10
D) 8
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