2016502858579884466176 - Numberphile
Audio Brief
Show transcript
This episode delves into the fascinating world of Harshad numbers, also known as Niven numbers, exploring their unique properties and characteristics.
There are four key takeaways from this conversation. Harshad numbers are integers divisible by the sum of their own digits. Sequences of consecutive Harshad numbers are limited to a maximum of 20. Only four numbers are 'universal joy givers,' being Harshad in every numerical base, and every integer under one billion is either a Harshad number or a sum of two.
A Harshad number, a term derived from the Sanskrit for 'joy giver,' is precisely defined as an integer that is perfectly divisible by the sum of its individual digits. For example, 21 is a Harshad number because the sum of its digits, 2 plus 1, equals 3, and 21 is divisible by 3.
While sequences of consecutive Harshad numbers do exist, mathematical proofs establish a definitive maximum. No sequence of integers that are all Harshad numbers can extend beyond 20.
An exceptionally rare category, 'trans Harshad' or 'universal joy giver' numbers are Harshad in every numerical base, not just base 10. Only four such numbers are known to exist: 1, 2, 4, and 6.
Further highlighting their widespread presence, it has been shown that every integer under one billion can be categorized. Each number is either a Harshad number on its own, or it can be represented as the sum of two distinct Harshad numbers.
This exploration into Harshad numbers reveals a world of unexpected mathematical elegance and surprising patterns.
Episode Overview
- The episode introduces the concept of Harshad numbers, also known as Niven numbers, through a puzzle involving the numbers 2025, 666, and 1729.
- A Harshad number is defined as any integer that is divisible by the sum of its digits. The name originates from the Sanskrit word harṣa, meaning "joy giver."
- The discussion explores various fascinating properties and categories of Harshad numbers, such as consecutive sequences, factorial relationships, and special types like "super harshad" and "trans-harshad" numbers.
- The presenter demonstrates how to check if a number is a Harshad number and even applies the test to host Brady Haran's birthday, with surprising results.
Key Concepts
- Harshad Number (or Niven Number): An integer that is divisible by the sum of its own digits. For example, 21 is a Harshad number because 2 + 1 = 3, and 21 is divisible by 3.
- Consecutive Harshad Numbers: There can be sequences of consecutive integers that are all Harshad numbers. However, it has been proven that no sequence of consecutive Harshad numbers can be longer than 20.
- Harshad Chains: A process where you start with a Harshad number, divide it by its digit sum to get another number, and then test if that new number is also a Harshad number, continuing the chain.
- Strong/Super Harshad Numbers: A Harshad number which, when divided by the sum of its digits, results in a prime number. For example, 18 is a strong Harshad number because 18 / (1+8) = 2, which is prime.
- Trans-Harshad Numbers (Universal Joy Givers): These are numbers that are Harshad in every numerical base. Only four such numbers exist: 1, 2, 4, and 6.
- Harshadmorphic Numbers: A number 'n' is harshadmorphic if there exists a Harshad number that ends in 'n' and also has a digit sum of 'n'.
Quotes
- At 00:17 - "They're Harshad numbers." - Brady Haran correctly guesses the topic of the video immediately, comically preempting the puzzle.
- At 00:24 - "They bring joy. These are numbers that bring joy." - The presenter explains that the name "Harshad" comes from a Sanskrit word meaning "joy giver."
- At 02:19 - "Brady, your birthday's rubbish." - After calculating that Brady Haran's birthday (18061976) is not a Harshad number, the presenter jokingly dismisses it.
Takeaways
- You can determine if any number is a "joy giver" by summing its digits and checking if the original number is divisible by that sum.
- There is a maximum limit to how many consecutive Harshad numbers can exist in a row, which is 20.
- Only four numbers are Harshad in every possible base (1, 2, 4, and 6), making them "universal joy givers."
- Every integer under one billion can be expressed as either a Harshad number itself or as the sum of two Harshad numbers.
