Number theory and arithmetic functions:
divisors d(n), sum of divisors σ(n), totient φ(n), prime factorization
• Prime factors calculator (up to 20 digits)
• Euler's totient function φ(n) (up to 20 digits)
• Divisors calculator: σ(n) and d(n) (up to 20 digits)
• Highly composite numbers (up to 20 digits)
• Big integer calculator: arbitrary precision arithmetic (10000+ digits)
• The Miller-Rabin primality test
• Firoozbakht conjecture:
(pk)k+1 > (pk+1)k
• Generalized Legendre conjecture:
Is there a prime between ns and (n+1)s for s < 2?
• Primes between squares:
How many primes are there between n2 and (n+1)2?
• Primes between cubes:
How many primes are there between n3 and (n+1)3?
• Stronger-than-Goldbach conjecture:
Large even numbers are sums of prime twins!?
Prime constellations or k-tuples
• Maximal gaps between prime k-tuples:
gk(p) < Mk logk+1p.
• Prime Twins
• Prime Quadruplets
• Prime Sextuplets
Probability and statistics
• Maximal intervals between rare random events
• City Distance Calculator
More on factorization
• Factorization of numbers up to 9007199254740992 = 253
• Finding the least factor in integers under 1020 (to be published soon)