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Amicable Number Checker

Check whether two positive integers form an amicable pair, or enter just one number and let the tool discover its partner automatically. Features animated divisor "handshake" visualizations, step-by-step sigma function breakdowns, aliquot-chain previews, and historical context dating back to Pythagoras.

Amicable Number Checker

Try a famous pair — or a single number to auto-discover its partner:

✦ Classic Pair 220, 284 1184, 1210 2620, 2924 17296, 18416
→ Auto Find 220 5020 12285
✕ Not Amicable 6, 28 100, 200
↔

↔ Leave the second field empty to auto-compute the candidate partner and verify the loop.

Embed Amicable Number Checker Widget

About Amicable Number Checker

Welcome to the Amicable Number Checker, an interactive tool that verifies whether two positive integers form an amicable pair — one of the most elegant relationships in number theory. You can enter a pair to verify, or provide a single number and let the tool discover its candidate partner automatically. The result page includes a five-step proof, a handshake diagram showing the two cross-sum conditions, a side-by-side divisor breakdown, and an aliquot-chain preview.

What Are Amicable Numbers?

Two distinct positive integers \(a\) and \(b\) form an amicable pair if the sum of the proper divisors of each one equals the other. In other words, the aliquot sum — the sum of all positive divisors of a number excluding itself — points from \(a\) to \(b\) and from \(b\) back to \(a\).

Definition of an Amicable Pair
$$s(a) = b \quad \text{and} \quad s(b) = a, \quad a \neq b$$

where \(s(n)\) is the sum of the proper divisors of \(n\)

The smallest amicable pair is (220, 284):

Each number "generates" the other through its own divisors — hence the name amicable (from Latin amicabilis, meaning friendly).

A Short History of Amicable Numbers

Amicable numbers have fascinated mathematicians for over 2,500 years:

How to Use This Calculator

  1. Enter numbers: Type one or two positive integers. Leave the second field blank to let the tool automatically find a candidate partner.
  2. Check: Click "Check Amicable Pair" to run the verification.
  3. Read the verdict: The colored banner at the top shows whether the pair is amicable (green) or not (red).
  4. Explore: Review the handshake diagram, side-by-side divisor breakdown, step-by-step proof, divisor bar charts, and aliquot chain preview.

Thabit ibn Qurra's Rule

Around 850 AD, the Arab polymath Thabit ibn Qurra found a partial formula for generating amicable pairs. Let:

Thabit's Rule
$$p = 3 \cdot 2^{n-1} - 1, \quad q = 3 \cdot 2^{n} - 1, \quad r = 9 \cdot 2^{2n-1} - 1$$

If \(p, q, r\) are all prime, then \(\left(2^n \cdot p \cdot q, \; 2^n \cdot r\right)\) is an amicable pair.

Setting \(n = 2\) gives \(p=5, q=11, r=71\) — all prime — producing the classical pair (220, 284). The rule yields valid results only for a handful of \(n\) values and is therefore not exhaustive, but it gave mathematicians a foothold for finding new pairs centuries before computers.

Aliquot Sequences & Sociable Numbers

The aliquot sequence of a number \(n\) is the sequence \(n, s(n), s(s(n)), \ldots\) obtained by repeatedly applying the proper divisor sum. What a sequence does reveals deep structure:

First Ten Amicable Pairs

#SmallerLargerDiscovered by
1220284Pythagoras (c. 500 BC)
21,1841,210Paganini (1866)
32,6202,924Euler (1747)
45,0205,564Euler
56,2326,368Euler
610,74410,856Euler
712,28514,595Brown (1939) — smallest odd pair
817,29618,416Ibn al-Banna / Fermat
963,02076,084Euler
1066,92866,992Euler

Fun Facts About Amicable Numbers

Frequently Asked Questions

What are amicable numbers?

Amicable numbers are two distinct positive integers (a, b) such that the sum of the proper divisors of a equals b, and the sum of the proper divisors of b equals a. The smallest amicable pair is (220, 284), attributed to Pythagoras.

How do I check whether two numbers are amicable?

Compute the proper divisors (all divisors less than the number itself) of both numbers and sum them. If \(s(a) = b\) and \(s(b) = a\), and \(a \neq b\), then \((a, b)\) is an amicable pair. Our tool does this automatically and shows each step.

Can I enter just one number to find its amicable partner?

Yes. Leave the second field blank and the tool will compute \(s(a)\) as the candidate partner, then check whether \(s(s(a)) = a\). If it does, the two numbers form an amicable pair.

What is the difference between amicable and perfect numbers?

A perfect number is a single number that equals the sum of its own proper divisors (e.g., 6 = 1+2+3). An amicable pair consists of two distinct numbers where each equals the sum of the other's proper divisors. Perfect numbers can be seen as the degenerate case where \(a = b\), but by convention they are not called amicable.

How many amicable pairs are known?

As of the 2020s, more than 1.2 billion amicable pairs have been computed by collaborative projects. The first few are (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), and (6232, 6368). The smallest known odd amicable pair is (12285, 14595).

Who discovered the pair (1184, 1210)?

It was found in 1866 by Niccolò Paganini, a 16-year-old Italian student. This pair was overlooked by centuries of mathematicians including Fermat, Descartes, and Euler, despite being the second-smallest amicable pair.

Additional Resources

Reference this content, page, or tool as:

"Amicable Number Checker" at https://MiniWebtool.com/amicable-number-checker/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 18, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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