mathematics

Thabit number

Thabit number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Thabit number rather than just read about it. In short: In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 ⋅ 2 n − 1 {\displaystyle 3\cdot 2^{n}-1} for a non-negative integer n. The first few Thabit numbers are: 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ...

Key takeaways

Thabit number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Thabit number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Thabit number from memory before moving on to harder problems.

Reference excerpt

2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... (sequence A055010 in the OEIS) The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers.

Properties The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes):

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... (sequence A007505 in the OEIS) As of June 2025, there are 69 known prime Thabit numbers. Their n values are:

0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718, 16819291, 17748034, 18196595, 18924988, 20928756, 22103376, 23157875, ... (sequence A002235 in the OEIS) The primes for 234760 ≤ n ≤ 3136255 were found by the distributed computing project 321 search. In 2008, PrimeGrid took over the search for Thabit primes. It is still searching and has already found all currently known Thabit primes with n ≥ 4235414. It is also searching for primes of the form 3·2n+1, such primes are called Thabit primes of the second kind or 321 primes of the second kind. The first few Thabit numbers of the second kind are:

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, ... (sequence A181565 in the OEIS) The first few Thabit primes of the second kind are:

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, ... (sequence A039687 in the OEIS) Their n values are:

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346, 16408818, ... (sequence A002253 in the OEIS)

Connection with amicable numbers When both n {\displaystyle n} and n − 1 {\displaystyle n-1} yield Thabit primes (of the first kind), and 9 ⋅ 2 2 n − 1 − 1 {\displaystyle 9\cdot 2^{2n-1}-1} is also prime, a pair of amicable numbers can be calculated as follows:

For example, n = 2 {\displaystyle n=2} gives the Thabit prime 11, n − 1 = 1 {\displaystyle n-1=1} gives the Thabit prime 5, and the third term is 71. Then, 22=4, multiplied by 5 and 11 results in 220, whose divisors add up to 284, and 4 times 71 is 284, whose divisors add up to 220. The only known values of n {\displaystyle n} satisfying these conditions are 2, 4 and 7, corresponding to the Thabit primes 11, 47 and 383 given by n {\displaystyle n} , the Thabit primes 5, 23 and 191 given by n − 1 {\displaystyle n-1} , and the third terms 71, 1151 and 73727. The corresponding amicable pairs are (220, 284), (17296, 18416) and (9363584, 9437056).

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Thabit number

Start with the simplest possible case. Write down what Thabit number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Thabit number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Thabit number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Thabit number

In research: Thabit number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Thabit number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Thabit number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arab inventions, Integer sequences, Mathematics in the medieval Islamic world, so understanding it makes those chapters shorter.
In everyday life: Look for Thabit number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Thabit number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Thabit number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Thabit number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Thabit number in simple terms?

In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 ⋅ 2 n − 1 {\displaystyle 3\cdot 2^{n}-1} for a non-negative integer n. The first few Thabit numbers are: 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 3…

Why does Thabit number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Thabit number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Thabit number.