Number Gossip
(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)
Unique Properties of 1729
- 1729 is a Ramanujan number, also called a taxi-cab number, the smallest number representable in two ways as a sum of two cubes: 1729 = 13+123 = 93+103
- 1729 is the smallest number that is a pseudoprime simultaneously to bases 2, 3 and 5
- Web pages about 1729:
- 1729 is the smallest Carmichael number n such that n-1 is a perfect power
- 1729 is the smallest Carmichael number whose prime factors form an arithmetic progression
- 1729 is the product of the first three happy primes
- 1729 is the smallest Carmichael number such that its Euler totient function is a perfect square
- 1729 is the smallest phi-radical number: composite number n such that rad(φ(n)) = rad(n-1)
Rare Properties of 1729
The composite integer n is a Carmichael number if bn-1 = 1 (mod n) for every integer b
which is relatively prime with n.
Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.
Common Properties of 1729
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
The number n is deficient if the sum of all its positive divisors except itself is less than n.
Compare with perfect and abundant numbers.
A number is odd if it is not divisible by 2.
Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.
The number n is odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
A number is said to be square-free if its prime decomposition contains no repeated factors.