Thabit number

In number theory, a Thabit number, also called a Thābit ibn Qurra number or 321 number, is a special kind of integer. It has a specific form: 3 times 2 raised to the power of n, then minus 1. Here, n is a non-negative integer, meaning it can be zero or any whole number greater than zero.

Some examples of Thabit numbers include 2, 5, 11, 23, 47, 95, and many more. These numbers appear in a list that goes on forever, and they are studied by mathematicians.

A wise man named Thābit ibn Qurra, who lived in the 9th century, was the first to look closely at these numbers. He was not just a mathematician; he was also a physician, astronomer, and translator. Thābit ibn Qurra discovered that Thabit numbers have an interesting connection to another group of special numbers called amicable numbers. His work helped people understand these patterns better.

Properties

Thabit numbers have a special pattern in their binary form. For a Thabit number of the form 3·2ⁿ−1, its binary representation has n+2 digits. It starts with "10" followed by n ones.

Some Thabit numbers are also prime numbers, known as Thabit primes or 321 primes. As of June 2025, there are 69 known Thabit primes. These primes are special because they come from specific values of n.

There is also a search for primes of a similar form, 3·2ⁿ+1, called Thabit primes of the second kind or 321 primes of the second kind. These have their own interesting patterns and values of n as well.

Connection with amicable numbers

When certain special numbers called Thabit primes are found in a specific way, they can help us discover pairs of numbers known as amicable numbers. These pairs have a unique property: the sum of the divisors of each number equals the other number.

For example, when we use the number 2, we find that the numbers 220 and 284 form an amicable pair. The divisors of 220 add up to 284, and the divisors of 284 add up to 220. This special relationship shows how Thabit numbers can be linked to these interesting pairs of numbers.

Generalization

A Thabit number base b is a special kind of number. It is made by using a formula: (b + 1) times b raised to the power of n, then subtract 1. Here, b must be at least 2, and n can be any whole number, including zero.

There are also Thabit numbers of the second kind base b. These are made in a similar way, but instead of subtracting 1, we add 1 to the result.

Other special numbers, called Williams numbers, are also based on similar formulas. For Williams numbers, we use (b − 1) instead of (b + 1) in the formula. There are also Williams numbers of the second kind, where we add 1 instead of subtracting 1.

Some of these numbers can be prime numbers, which are numbers that can only be divided by 1 and themselves. When a Thabit number base b is prime, it is called a Thabit prime base b. The same idea applies to Williams primes.

There are many ideas and guesses about how many of these special prime numbers exist, but these are still being studied by mathematicians.