A Brief History of Amicable Numbers

Circa 500 B.C., the Greeks recognized certain numbers as being particularly significant. Among these were two unique numbers, 220 and 284.

To grasp their significance, we must first define a proper divisor of a number ( n ) as any number ( d ) that divides ( n ) and is less than ( n ). Remarkably, the sum of the proper divisors of 220 equals 284, while the proper divisors of 284 sum up to 220.

If you wish to verify this, you can calculate that the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, giving us:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

For 284, the proper divisors are 1, 2, 4, 71, and 142, leading to:

1 + 2 + 4 + 71 + 142 = 220.

This unique relationship is termed amicability, and the numbers are referred to as amicable pairs (with amicable meaning friends or lovers). Notably, the pair (220, 284) stands as the earliest known amicable pair.

In ancient traditions, lovers would engage in a ritual involving fruit, inscribing one of these numbers on each half before sharing it, symbolizing their eternal bond. This practice was sometimes conducted discreetly for love that was forbidden or secretive, solidifying their connection in the realm of amicability.

Despite their fascination, the Greeks struggled to discover additional amicable pairs for nearly a millennium. It wasn't until the 9th century that Thabit ibn Qurra identified two more pairs.

During this period, the focus of mathematics transitioned from Europe and Egypt to the Arabic world, where it thrived for almost five centuries. Thabit's ingenious rule enabled him to generate new amicable numbers.

Unfortunately, his findings, along with advancements in mathematics from Iran, did not reach Europe, where only the original Greek pair was recognized until Pierre de Fermat found another pair in 1636: 17,296 and 18,416.

Fermat, a lawyer and amateur mathematician, was renowned for his early explorations in calculus and solving complex number theory problems. During this era, a mathematical rivalry brewed between Fermat and René Descartes, both of whom were giants in the field. Following Fermat's discovery, Descartes felt compelled to find another pair.

After two years of intense calculations, Descartes unveiled the pair 9,363,584 and 9,437,056 in 1638, all without the aid of a calculator.

Leonhard Euler

Interestingly, both pairs discovered by Fermat and Descartes were the same as those previously identified by Thabit, having been lost and then rediscovered.

Thus, for over 2000 years, only three amicable pairs were known, until Euler entered the scene in 1750.

In his work titled "De numeris amicabilibus," Euler uncovered 58 additional pairs of amicable numbers! As the esteemed writer Wilhelm Dunham noted: “The world supply of amicable numbers increased twentyfold in a single paper.”

This was no mere coincidence; Euler employed a method based on the sum-of-divisors function and some brilliant insights.

Euler thoroughly investigated the sum-of-divisors function in various papers, including those discussing perfect numbers—numbers that are self-amicable, meaning ( sigma(n) = 2n ). A classic example is 6, since ( sigma(6) = 1 + 2 + 3 + 6 = 12 = 2 times 6 ).

The sequence of perfect numbers begins with: 6, 28, 496, 8128, …

It turns out that the sum-of-divisors function ( sigma ) is multiplicative. If ( n ) and ( m ) are natural numbers without common prime factors, then ( sigma(nm) = sigma(n)sigma(m) ). Furthermore, if two numbers ( n ) and ( m ) are amicable, it follows that ( sigma(n) - n = sigma(m) - m ) or alternatively, ( sigma(n) = n + m = sigma(m) ).

By manipulating this equation with specific multiplicative forms of ( n ) and ( m ) possessing particular prime factors, he devised a strategy to generate nearly 60 new amicable pairs!

Modern Discoveries

In 1867, a remarkable discovery was made by a 16-year-old schoolboy named B. Nicolò I. Paganini, who uncovered the second smallest amicable pair, (1184, 1210). Astonishingly, this pair had evaded the attention of the greatest mathematicians for over two millennia, including Euler.

Today, with the power of computers, we can explore the gaps between known pairs and have discovered more than a billion amicable pairs. Despite these advancements, our understanding of amicability has made little progress since Euler's time.

Today, we remain uncertain about the infinite nature of amicable or perfect numbers, and the existence of a solitary odd perfect number remains a mystery. It is somewhat disheartening given the extensive exploration over the last 2500 years.

But why does this matter? While it may not directly contribute to advancements in quantum computing or enable faster-than-light travel, the pursuit of understanding these numbers can deepen our knowledge of the sum-of-divisors function, which is intricately linked to various significant mathematical theories, such as the Riemann hypothesis.

Any fresh insight into the sum-of-divisors function would be an extraordinary achievement in its own right.

This first video titled "Angel Numbers for Love Marriage | Long-Term Relationship, Find a Soulmate" explores how numerical patterns can influence romantic connections and relationships.

The second video, "Love By The Numbers," delves into the intriguing relationship between numbers and the concept of love, further emphasizing the historical significance of amicable numbers.