10 Numbers With Surprisingly Interesting Properties
In the TV show The Big Bang Theory, the character Sheldon Cooper declared 73 "the best number." His reasons: it's the 21st prime, its mirror 37 is the 12th prime, and 21 is 7 × 3. Plus, in binary, 73 is a palindrome: 1001001.
It sounded like a throwaway joke. In 2019, mathematicians Carl Pomerance and Chris Spicer published a proof that 73 is the only number with exactly those properties.
It's a fun example of a serious idea: almost every number, looked at closely, has something remarkable about it. Here are 10 of the best.
There Are No Uninteresting Numbers
Here's a classic joke proof. Suppose some whole numbers are "uninteresting." Then there must be a smallest uninteresting number. But being the smallest uninteresting number is itself interesting. Contradiction! So every number is interesting.
It's tongue-in-cheek (the word "interesting" isn't mathematically defined), but the list below shows how easy it is to find something special.
1. 2: The Oddest Prime
2 is the only even prime. Every other even number is divisible by 2. This single exception shows up everywhere in number theory, which is why many theorems begin "for every odd prime p." See 2.
2. 6: The First Perfect Number
6 = 1 + 2 + 3, the sum of its proper divisors. It's also the product of those same divisors: 1 × 2 × 3 = 6. And it's the smallest number that's the product of two distinct primes. See Perfect Numbers.
3. 26: Sandwiched Between a Square and a Cube
25 = 5² 26 27 = 3³
26 is the only whole number that sits directly between a perfect square and a perfect cube. Pierre de Fermat is credited with proving it, and he reportedly challenged English mathematicians to prove it too. See 26.
4. 73: The "Sheldon Prime"
As above: 73 is the 21st prime, its reversal 37 is the 12th prime (12 reversed is 21), 7 × 3 = 21, and its binary form 1001001 reads the same both ways. Pomerance and Spicer proved no other number shares all of these properties. See 73.
5. 153: The Narcissistic Triangular Number
1³ + 5³ + 3³ = 153
153 equals the sum of the cubes of its digits. It's also the 17th triangular number: 1 + 2 + … + 17 = 153. See 153 and Narcissistic Numbers.
6. 1729: The Taxicab Number
1729 = 1³ + 12³ = 9³ + 10³
The smallest number that's a sum of two cubes in two different ways, as Srinivasa Ramanujan pointed out to G. H. Hardy about a taxicab's number. See 1729.
7. 2520: Divisible by Everything Up to 10
2520 is the smallest number divisible by every number from 1 to 10. It's the least common multiple of 1 through 10:
2520 = 2³ × 3² × 5 × 7
Check it with the LCM calculator and see 2520.
8. 6174: Kaprekar's Constant
Take almost any four-digit number, arrange its digits from largest to smallest and smallest to largest, and subtract. Repeat. You'll reach 6174 within seven steps, and stay there, since 7641 − 1467 = 6174. See Why 6174 Is Called the Kaprekar Constant.
9. 142857: The Cyclic Number
Multiply 142857 by 1 through 6:
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
Every product uses the same digits in the same cyclic order. Then:
142857 × 7 = 999999
The reason: 1/7 = 0.142857142857… The number is the repeating block of 1/7, and multiplying by 2 through 6 just shifts where the cycle starts. See 142857.
10. 196: The Stubborn Palindrome Candidate
Reverse the digits of most numbers and add, and you'll quickly get a palindrome. 196 has been pushed through hundreds of millions of digits without ever producing one, yet nobody has proved that it never will. See 196.
An Insider Reference: The Number Collector
Many of these facts are catalogued in the On-Line Encyclopedia of Integer Sequences (OEIS), founded by mathematician Neil Sloane. He began collecting sequences on index cards in 1964 as a graduate student, published books of them in 1973 and 1995, and put the collection online in 1996.
The OEIS now contains hundreds of thousands of sequences and is one of the most cited resources in mathematics. Researchers regularly discover that a sequence arising in their work already appears there, sometimes from a completely unrelated field.
Two Concepts Worth Knowing
Least Common Multiple
The least common multiple of several numbers is the smallest number divisible by all of them. Take the highest power of each prime that appears in any factorization. See Real-Life Uses of LCM and GCF.
Repeating Decimal
Every fraction has a decimal that terminates or eventually repeats. The repeating block of 1/p for certain primes p produces cyclic numbers like 142857.
Quick Answer: What Numbers Have Interesting Properties?
Examples include 2 (the only even prime), 6 (the first perfect number), 26 (the only number between a square and a cube), 73 (the "Sheldon prime"), 153 (equal to the sum of the cubes of its digits), 1729 (the taxicab number), 2520 (smallest number divisible by 1–10), 6174 (Kaprekar's constant) and 142857 (a cyclic number from 1/7).
Try Them Yourself
- Number 142857: the cyclic number
- Number 2520: divisible by 1 through 10
- Number 73: the Sheldon prime
- LCM Calculator: find the smallest number divisible by 1 through 12
- Number of the Day: a new interesting number every day
- Significant Numbers and Their Meanings: numbers with cultural significance
Look up your birth year or house number on Math Tools. Find one property that makes it special. The "no uninteresting numbers" joke says you're guaranteed to succeed.