Amicable Number Checker

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 \(a\).

The smallest amicable pair is (220, 284):

• Proper divisors of 220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
• Proper divisors of 284: 1 + 2 + 4 + 71 + 142 = 220

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:

• Pythagoras (c. 500 BC): According to Iamblichus, Pythagoras knew the pair (220, 284) and called it a symbol of friendship.
• Thabit ibn Qurra (9th century): Discovered the first general rule for generating amicable pairs — now known as Thabit's theorem.
• Ibn al-Banna (13th century): Discovered the pair (17296, 18416), rediscovered by Fermat in 1636.
• Fermat & Descartes (17th century): Independently found (9,363,584; 9,437,056).
• Euler (18th century): Vastly expanded the list, discovering 59 new pairs and formalizing the theory.
• Paganini (1866): A 16-year-old Italian named Niccolò Paganini found (1184, 1210), the second-smallest pair — which every great mathematician before him had missed.
• Modern era: As of the 2020s, collaborative computation has found more than 1.2 billion amicable pairs.

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:

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:

• Perfect numbers form fixed points: \(s(n) = n\) (period 1).
• Amicable pairs form 2-cycles: \(s(s(n)) = n\) (period 2).
• Sociable numbers form longer cycles of period 3 or more (e.g., the 5-cycle starting at 12496).
• Aspiring numbers eventually reach a perfect number.
• Deficient chains drop to 1 and terminate.
• Lehmer five: the sequences starting at 276, 552, 564, 660, and 966 have been computed to billions of terms without resolution — their fate is unknown.

First Ten Amicable Pairs

Fun Facts About Amicable Numbers

• In the Bible, Jacob offers Esau 220 goats as a peace gift (Genesis 32:14) — some scholars see a nod to the amicable pair (220, 284).
• Medieval talismans occasionally engraved 220 and 284 on two objects exchanged between friends or lovers.
• All known amicable pairs share the same parity: both even or both odd — no mixed-parity pair has ever been found, though whether any could exist is an open problem.
• Every known amicable pair also shares a common factor greater than 1. Whether a coprime amicable pair exists remains unsolved, and if one exists, it must exceed \(10^{67}\).

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

• Amicable Numbers - Wikipedia
• Aliquot Sequence - Wikipedia
• Sociable Numbers - Wikipedia
• Thabit ibn Qurra's Rule - Wikipedia
• OEIS A259180: Amicable pairs

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