Amicable Numbers Research Articles

Reduced and Augmented Amicable Pairs to 10 8

Reduced and Augmented Amicable Pairs to 10 ⁸

A Search for Aliquot Cycles and Amicable Pairs

A search for aliquot cycles below $3.6 \cdot {10^{10}}$ and amicable pairs below ${10^{11}}$ is described. Three new cycles of length 4 and one new cycle of length 6 are exhibited. Four triples of amicable pairs with the same pair-sum are also exhibited.

A List of Amicable Pairs Below 10 11 .

A List of Amicable Pairs Below 10 11 .

A search for aliquot cycles and amicable pairs

A search for aliquot cycles below 3.6 ⋅ 10 10 3.6 \cdot {10^{10}} and amicable pairs below 10 11 {10^{11}} is described. Three new cycles of length 4 and one new cycle of length 6 are exhibited. Four triples of amicable pairs with the same pair-sum are also exhibited.

Two problems of number theory in Islamic times

Let o{n) denote the sum of the positive divisors of the integer n and s(n) the sum of its aliquot parts, that is, of the divisors of n other than n itself; then, s(n) =■■ o{ri) n. Greek mathematicians had considered the problem of the determination of a perfect number (teXeloc dpt^oc), that is, a number n such that s(n) = n. Euclid demonstrated that if 2m 1 is a prime, then 2"I~I(2m 1) is perfect. Euler proved that any even perfect number must have this form, so that the question of the generality of Euclid's formula depends upon the existence of odd perfect numbers. This remains an open question today since no odd perfect number has been found but no demonstration of their non-existence has been found either. The idea of considering these perfect numbers goes back to the Pythagoreans. Such is also the case for the so-called amicable numbers (ciXoi apiifyoi), that is, pairs of numbers at,, n2 such that each is the sum of the aliquot parts of the other, i.e., s(ni) = n2 and s(n2) = nA or (t(az,) v{n2) =:z nx + n2. The Pythagoreans apparently only knew of the pair 220,284, and a method for finding further pairs (actually having the very restricted form nx = 2hp, n2 2hqr, with /?, q, r different primes) was not discovered before the ninth century, by Thabit ibn Qurra. His rule was rediscovered in seventeenth century Europe, from which time research went further. Thus, Euler determined some sixty pairs, but not subject to the former restriction on their prime factorization. Two further questions in relation to the sums of divisors arose during Islamic times:

Amicable Pairs of the Form (i, 1)

Amicable Pairs of the Form (i, 1)

A search for aliquot cycles below 10¹⁰

A search for aliquot k-cycles below 10 10 {10^{10}} with k ≥ 3 k \geq 3 is described. Two new 4-cycles are exhibited. Six new 4-cycles not below 10 10 {10^{10}} are also exhibited.

Amicable pairs of the form (𝑖,1)

A method is given for finding amicable pairs of a certain type. When implemented, thirteen new amicable pairs were discovered. Using methods for finding new pairs from known pairs, the thirteen new pairs generated 64 other new pairs.

On an integer’s infinitary divisors

The notions of unitary divisor and biunitary divisor are extended in a natural fashion to give k-ary divisors, for any natural number k. We show that we may sensibly allow k to increase indefinitely, and this leads to infinitary divisors. The infinitary divisors of an integer are described in full, and applications to the obvious analogues of the classical perfect and amicable numbers and aliquot sequences are given.

Notes on Th [formula omitted]bit ibn Qurra and his rule for amicable numbers

On the basis of the Arabic text, we investigate how Th a bit ibn Qurra (ninth century) could have found his rule for amicable numbers.

Operations on Generators of Unitary Amicable Pairs

Operations on Generators of Unitary Amicable Pairs

The First Seven Perfect Numbers and Three Types of Amicable Numbers in a Manuscript on Elementary Number Theory by Ibn Fallûs

The First Seven Perfect Numbers and Three Types of Amicable Numbers in a Manuscript on Elementary Number Theory by Ibn Fallûs

Are There Odd Amicable Numbers Not Divisible by Three

Are There Odd Amicable Numbers Not Divisible by Three

Are there odd amicable numbers not divisible by three?

A conjecture of Bratley and McKay, according to which odd amicable numbers should be divisible by three, is disproved by some counterexamples.

Sums of Divisors, Perfect Numbers and Factoring

Let N be a positive integer, and let $\sigma (N)$ denote the sum of the divisors of N (e.g. $\sigma (6) = 1 + 2 + 3 + 6 = 12$). We show computing $\sigma (N)$ is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given $\sigma (N)$, produces the prime factorization of N, and $\sigma (N)$ can be computed in polynomial time given the factorization of N.We show that the same result holds for $\sigma _k (N)$, the sum of the kth powers of divisors of NWe give three new examples of problems that are in Gill’s complexity class BPP: perfect numbers, multiply perfect numbers, and amicable pairs. These are the first “natural” sets in BPP that are not obviously in RP.

Computation of All the Amicable Pairs Below 10 10

An efficient exhaustive numerical search method for amicable pairs is .described. With the aid of this method all 1427 amicable pairs with smaller member below 1010 have been computed, more than 800 pairs being new. This extends previous exhaustive work below 108 by H. Cohen. In three appendices (contained in the supplements section of this issue), various statistics are given, including an ordered list of all the gcd's of the 1427 amicable pairs below 1010 (which may be useful in further amicable pair research). Suggested by the numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.

Supplement to Computation of All the Amicable Pairs Below 10 1 0

Supplement to Computation of All the Amicable Pairs Below 10 1 0

Computation of all the amicable pairs below 10¹⁰

An efficient exhaustive numerical search method for amicable pairs is described. With the aid of this method all 1427 amicable pairs with smaller member below 10 10 {10^{10}} have been computed, more than 800 pairs being new. This extends previous exhaustive work below 10 8 {10^8} by H. Cohen. In three appendices (contained in the supplements section of this issue), various statistics are given, including an ordered list of all the gcd’s of the 1427 amicable pairs below 10 10 {10^{10}} (which may be useful in further amicable pair research). Suggested by the numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.

Breeding Amicable Numbers in Abundance

In a first article of this title, new procedures were described to compute many amicable numbers by breeding them in several generations. An extensive computer search was later performed (in 1988), and demonstrated the remarkable effectiveness of this breeding method: the number of known amicable pairs was easily quadrupled by this search. As we learnt recently (1999) from the internet, Pederson and te Riele have again multiplied that number roughly by ten. While they give no information on their method of search, we publish here our method and summarize the computations. Our results provide some evidence for the conjecture that the number of amicable pairs is infinite.

Breeding amicable numbers in abundance

We give some new methods for the constructive search for amicable number pairs. Our numerical experiments using these methods produced a total of 3501 new amicable pairs of a very special form. They provide some experimental evidence for the infinity of such pairs.