Number Facts Explorer
Everything about any number up to a billion — prime factorisation, divisors, binary, Roman numerals, and the properties that make some numbers unusual.
The first number with nothing to say for itself is 14
There is an old joke that no number can be uninteresting, because the smallest uninteresting one would be interesting for exactly that reason. Made concrete, it stops being a paradox and becomes a countable question. Take 13 properties a number either has or does not — prime, a perfect square, a perfect cube, a triangular number, a palindrome, a Fibonacci number, and seven more — and check every number in turn. Each of 1 through 13 has at least one. 14 has none. Its neighbours are both spoken for: 13 is prime and a Fibonacci number, and 15 is a triangular number.
That answer belongs to the list, not to arithmetic — admit Catalan numbers and 14 is rescued, since it is the fourth of them. What does not depend on the list is what happens as the numbers grow.
| Range | Have a property | Excluding primes |
|---|---|---|
| 1 – 10 | 100.0% | 100.0% |
| 1 – 100 | 65.0% | 47.0% |
| 101 – 1,000 | 33.0% | 18.9% |
| 1,001 – 10,000 | 15.5% | 3.8% |
| 10,001 – 100,000 | 11.0% | 1.8% |
The joke works on small numbers because down there the properties are crowded on top of each other. Squares sit 4 apart near 4 and 400 apart near 40,000; there are four factorials below 25 and eight below 100,000. By five digits, roughly nine numbers in ten have nothing on the list at all. Interestingness is not a property of numbers so much as a property of being near the start of them.
And past a few hundred, interesting mostly means prime
The third column above is the same count with primes struck out, and it falls much faster. Among the first hundred numbers, removing primes costs little — the small numbers are so thoroughly covered by everything else that the primes are mostly redundant. Among five-digit numbers, 98.2% have nothing but their primality to offer, and 89% have nothing at all.
The reason is how fast each kind thins. Primes thin logarithmically, so there are still 8,363 of them between 10,000 and 100,000. Squares thin like a square root, cubes worse, and factorials worse again. Every other property on the list runs out; primality does not.
Going the other way, the most decorated numbers are all tiny:
| Number | Properties | Which |
|---|---|---|
| 1 | 9 | a perfect square, a perfect cube, a triangular number, a palindrome, a Fibonacci number, a factorial, a power of two, a repdigit, highly composite |
| 2 | 8 | prime, a palindrome, a Fibonacci number, a factorial, a power of two, a repdigit, highly composite, a pronic number |
| 6 | 7 | a triangular number, a palindrome, a factorial, a perfect number, a repdigit, highly composite, a pronic number |
| 4 | 6 | a perfect square, a palindrome, a power of two, a repdigit, highly composite, a perfect power |
| 8 | 6 | a perfect cube, a palindrome, a Fibonacci number, a power of two, a repdigit, a perfect power |
| 3 | 5 | prime, a triangular number, a palindrome, a Fibonacci number, a repdigit |
1 carries 9 of the 13 at once. Nothing above 10,000 carries more than four. The famous small numbers are famous because they had almost no competition.
How to use
- Enter any whole number.
- Read its factorisation, divisors and representations.
- Look at the properties it does and does not have.
- Try a highly composite number to see the divisor count jump.
Frequently asked questions
What is a perfect number?
One equal to the sum of its proper divisors — 6 is 1 plus 2 plus 3. They are rare: only a handful are known below a trillion, all even, and whether any odd perfect number exists is an open problem over two thousand years old.
What makes a number highly composite?
Having more divisors than any smaller number. 12, 24, 36 and 60 are examples, which is precisely why they turn up in timekeeping, angle measurement and old currency systems — plenty of divisors makes a unit easy to divide into halves, thirds and quarters without fractions.
What is a triangular number?
The sum of consecutive whole numbers from 1 up, so 1, 3, 6, 10 and so on — the count of objects arranged in a triangle. Gauss famously computed such a sum as a schoolboy by pairing the ends, which is the standard proof of the formula.
Why does 1729 keep appearing?
The taxicab number, from Ramanujan's remark to Hardy that it is the smallest number expressible as a sum of two cubes in two different ways. Hardy had described the number as dull; the exchange is one of the better-known anecdotes in mathematics.
What is a palindromic number?
One reading the same in both directions, such as 12321. Whether repeatedly reversing and adding always eventually produces a palindrome is unresolved — 196 is the smallest number for which it has never been shown to happen despite enormous computation.
Why is the factorisation slow for some numbers?
Because factoring is genuinely hard for large numbers with two large prime factors, which is the basis of RSA encryption. Numbers with small factors are found quickly; a semiprime made of two large primes resists every known classical method, and that resistance is what secures a great deal of internet traffic.
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